Algebra Course Research Articles

Validity Analysis of Development of Socrates-Based Linear Aljebra E-Modules

The aim of the study was to investigate the validity level of Socrates-based linear algebra e-module, both material validity and design. This is a research and development (R&D) with ADDIE procedure: analysis, design, develop, implement, and evaluate. The participants were 30 students and 2 lecturers who supervised linear algebra courses in the mathematics education study program of Universitas Muhammadiyah Metro. Data collection techniques uses questionnaires provided to material and design validators. Data analysis uses quantitative descriptive. The results showed that the validity rate xof Socrates-based linear algebra e-modules on material aspects was 81.33% with highly valid categories and on design aspects 88% with highly valid categories. Based on the results of the study obtained the conclusion that the e-module algebra linear based on Socrates is declared very valid. Furthermore, further research can be conducted related to the use of Socrates-based linear algebra e-modules in learning to find out effectiveness of using e-modules.

Analysis of tutors’ responses to students’ queries in a second linear algebra course at a mathematics support center

This paper analyses six years of tutor feedback produced after inquiries made by students in a second linear algebra course at a university mathematics support center (MSC). We utilized Mason’s (2002) pedagogical tactics to build a model to analyze MSC tutors' feedback responding to these students’ queries. The aim of this research was to investigate the nature of students’ difficulties with concepts in a second linear algebra course that emphasizes theories and proof, in addition to examining the tactics employed by tutors to resolve student difficulties. We analyzed 227 feedback comments from 44 tutors based on their interactions with 82 students over six years. Our findings indicated that the most common areas of difficulty were basis, vector space, subspace, span, and proof. Tutor tactics deployed included ‘being mathematical’, ‘simplifying and complexifying’, and ‘worked examples’. We also discuss some implications for linear algebra tutor training.

PERSEPSI MAHASISWA TERHADAP PENGGUNAAN APLIKASI GOOGLE MEET DAN GOOGLE FORM DI MASA PANDEMI

The purpose of this study is to describe students' perceptions of the use of Google Meet and Google Form.The research method used is descriptive using a survey method. Purposive sampling was used as a data collection technique. Data collection was carried out online using a questionnaire instrument and interviews with a sample of 34 students of the S1 Mathematics Education Study Program FKIP Unissula who used the Google Meet and Google Form applications when conducting lectures on algebra courses. The data collection technique is using Google Form, then data collection is carried out using the WhatsApp application using an unstructured interview instrument.The results showed that students' perceptions of the use of Google Meet and Google Form were classified as good criteria. This is because there are still obstacles in both applications. The constraint of the Google Meet application is the signal factor, an unstable signal will make users automatically leave Google Meet, while the problem with Google Form itself is the absence of math equation facilities. The hope is that there will be developments in the Google Meet and Google Form applications so that they can provide more comfort and convenience for their users.

Using guided notes to support learning in first-year calculus

This research explores the use of guided notes in a post-secondary calculus course and the extent to which use enhanced students’ success in a first year calculus course. Guided notes have been previously shown to be perceived as helpful to learning by students, and one study showed that the use of guided notes improved pass rates in a first year college algebra course. Using an experimental design, our results show that guided notes improved outcomes for students when compared to a control group that did not use guided notes–logistic regression modelling found evidence that guided notes, mathematical aptitude and gender (male) were associated with a higher probability of a student successfully completing the course. No significant interactions between guided notes and gender was observed. Implications for further research and practitioners are included.

Gender, prior knowledge, and the impact of a flipped linear algebra course for engineers over multiple years

AbstractBackgroundResearch shows that active pedagogies could play an important role in achieving more equitable outcomes for diverse groups of students in Science, Technology, Engineering, and Mathematics (STEM). Although flipped classes are a popular active methodology, there is a lack of high‐quality studies assessing their impact in ecologically valid settings and exploring how outcomes are related to gender and to prior education.PurposeThis paper presents two modified replications of an experimental study investigating the impact of the flipped class approach on students' achievement in a large, first‐year class in an engineering bachelor's degree.MethodologyWe added a new strand, progressively flipped over 3 years, to eight parallel strands of a high‐stakes mandatory linear algebra course for engineers. The study followed a replicated‐between‐subjects design, with students in the flipped strand learning the same material as those in the other strands and taking the same final exam.ResultsOur results demonstrate that the flipped format did not have any significant impact on students' achievement compared to traditional lecturing. However, both replications in the flipped condition show a reduced attainment gap for women and students with less prior knowledge in mathematics.ConclusionWhile the flipped class seems to have weaker effects on learning than other active methodologies, the evidence in this study indicates that it may have an impact on reducing the attainment gap between different groups of students. It may therefore be particularly interesting to consider in efforts to achieve more equitable outcomes for women and where students have heterogeneous educational backgrounds.

The Effectiveness of the Link2Success Program on Freshman Level Math Course Achievement

<p style="text-align:justify">To improve college students’ achievement and success rate in the College Algebra course, a new program called Link2Success (L2S) was implemented in several sections of the course at the study’s university. The program required students to increase their class time to six hours. Three of those hours were spent with certified tutors who helped students with the content preview and homework preparation. The purpose of this study was to investigate the effectiveness of the Link2Success program based on the achievement of College Algebra students. One College Algebra class with embedded L2S and one College Algebra class without L2S were randomly chosen and the grades of several assignments and exams were compared. A survey was given to L2S students to explore their experience and opinion about the program. Another survey was given to non-L2S students to find out if they felt that learning with an L2S program was beneficial to them and would have improved their grades. The results showed that L2S did not have a positive effect on the College Algebra students’ achievement. However, L2S students felt more confident and rated the program highly where as non-L2S students felt they would have benefited from the program if it were implemented in their class.</p>

Discovery learning in an interdisciplinary course on finite fields and applications

The authors describe their approach to teaching a course on finite fields and combinatorial applications, including block designs and error-correcting codes, using a hybrid of lectures and active learning. Under the discussed classroom model, there are two lecture days and two discovery-based discussion days each week. Discussions center around group activities that build intuition for abstract concepts while avoiding excessive technical machinery. Teaching this course presents some unique challenges, as much of the content typically appears in a second course in abstract algebra, yet the students exhibit a wide range of mathematical preparation.

Changes made to the teaching of linear algebra and calculus courses in the UK in response to the COVID-19 pandemic

In response to the COVID-19 pandemic, university mathematics departments in the UK adapted their teaching for 2020-21, with some courses being delivered digitally and others through a mixture of on-campus and digital delivery. A survey of linear algebra and calculus lecturers was carried out in the spring of 2021 to investigate what changes were made to courses, as well as lecturers’ perceptions of institutional decision making and support. This survey found that a majority of the 41 participants were satisfied that the choice of delivery mode was correct, although views about the importance of offering on-campus classes were mixed. There was a significant increase in the use of video clips made by the lecturer, video-conferencing software, discussion forums, electronic submission of written work and on-screen marking tools. Most lecturers reported a reduction in the amount of time that students were expected to be taking part in live teaching activities and an increase in the amount of time they were expected to be working on asynchronous activities. While some were keen to return to their previous practice, others were enthusiastic about retaining features introduced in response to the COVID-19 pandemic.

The Interpretations of the Mathematics Teachers Taking Post-Graduate Education to the Expression “You can’t add apples and pears”

The study aims to explore how the mathematics teachers taking post-graduate education use their Algebra knowledge for the expression, “You can’t add apples and pears” that is expressed in almost every segment of the society, in creating mathematical content. The data of the research consisted of the written answers of the thirty-seven (37) teachers, who had taken the Algebra Teaching course, to the question “There is a rule in mathematics as everyone says; explain the statement ‘You can’t add apples and pears’ mathematically” in the exams from the 2008-2009 academic year to, including 2015-2016 academic year. The participants took the algebra education in the courses named Algebra and Introduction to Algebra courses and sufficient algebra education was provided to answer the question in the courses. The document analysis method was employed in the study and the data were analyzed with the descriptive content analysis. The results indicated that most of the teachers were aware of their algebra knowledge, the number and quality of the mathematical contents in the answers were at a satisfactory level, but the visuals used in some contents were not appropriate.

Pedagogical Strategies to Enhance Learning in a Linear Algebra Course

In this study, we evaluate an innovative method to improve the teaching of linear algebra. The use of peer instruction in conjunction with a seminar strategy and supported by didactic engineering is proposed as a means to facilitate the mastery of abstract concepts, the undertaking of innovative research in problem solving, and the practical application of educational concepts. To assess the method’s effectiveness, the Students’ Evaluation of Educational Quality (SEEQ) questionnaire was administered and assessed. In addition, to determine whether this new teaching approach leads to better learning results than the traditional method, a comparison between control and experimental groups was conducted. The results indicate that the mixed method creates a more dynamic learning environment for students than the traditional method, contributes to students’ development of social interaction skills and learning process. In sum, the proposed method has been shown to enhance student motivation and reflection, which are critical to a productive, collaborative learning environment that directly impacts academic retention and performance.

What Group Theory Can Do for You: From Magmas to Abstract Thinking in School Mathematics

In this paper, we present a novel concept for an abstract algebra course at the secondary school level. Against the backdrop of findings from mathematics education research, we discuss benefits that arise from teaching abstract algebra to secondary school students. We derive suitable content items as well as educational ideas on how to implement them into mathematics classrooms. We build on frameworks from mathematics education research to design a teaching-learning sequence that enables a beneficial learning trajectory across three units—the result is a coherent teaching concept that encompasses Magmas in a hands-on way. It can be used for either extra-curricular activities or advanced courses in mathematics classes, and can be of benefit for mathematics learners.

The Fundamental Theorem of Finite Fields: A Proof from First Principles

A mathematics student’s first introduction to the fundamental theorem of finite fields (FTFF) often occurs in an advanced abstract algebra course and invokes the power of Galois theory to prove it. Yet, the combinatorial and algebraic coding theory applications of finite fields can show up early on for students in STEM. To make the FTFF more accessible to students lacking exposure to Galois theory, we provide a proof from algebraic “first principles.”

LU and CR Elimination

The reduced row echelon form ${\bf rref}(A)$ has traditionally been used for classroom examples: small matrices $A$ with integer entries and low rank $r$. This paper creates a column-row rank-revealing factorization ${A=CR}$, with the first $r$ independent columns of $A$ in $C$ and the $r$ nonzero rows of ${\bf rref}(A)$ in $R$. We want to reimagine the start of a linear algebra course by helping students to see the independent columns of $A$ and the rank and the column space. If $B$ contains the first $r$ independent rows of $A$, then those rows of ${A=CR}$ produce ${B=WR}$. The $r$ by $r$ matrix $W$ has full rank $r$, where $B$ meets $C$. Then the triple factorization ${A=CW^{-1}B}$ treats columns and rows of $A$ ($C$ and $B$) in the same way.

ANALYSIS OF STUDENTS' CREATIVE THINKING SKILLS IN LINEAR ALGEBRA COURSES DURING THE COVID-19 PANDEMIC

The purpose of this study is to analyze students' creative thinking skills in linear algebra courses in the Covid-19 period. This research was conducted in students of the third semester of the academic year 2021/2022 using 28 respondents. Data collection is done using a creative thinking questionnaire of 30 statement items that have been compiled according to indicators of creative thinking ability. The scale used in this study is the Likert scale. Data analysis to find out the ability to creatively think students used narrative analysis techniques with ideal criteria, namely ideal mean and ideal deviation standards. Based on data analysis obtained as a result of using indicators: fluency (72.50), elaboration (78.00), Flexibility (72.00), originality (70.00). So it can be concluded that the creative thinking ability of students in linear algebra courses is in a moderate category with an average score of 73.13.

PROFIL KEMAMPUAN PEMBUKTIAN MATEMATIS MAHASISWA DALAM MATA KULIAH ALJABAR ABSTRAK

This research aims to examine the profile of the mathematical proving ability of students in Abstract Algebra courses. The method of this research is included in this type of descriptive research. The subjects of this study were 3 students who studied Abstract Algebra courses from the high, medium and low mathematical proofing ability group. The results showed that students with high ability and are having the ability to read mathematical proofs and perform mathematical proofs. As for the ability to criticize proof by adding, reducing or reconstituted a mathematical proof for each group of students is still not visible.

АЛГЕБРАИЧЕСКИЕ УРАВНЕНИЯ В УНИТАРНОМ ПРОСТРАНСТВЕ И КРАТЧАЙШЕЕ АЛГЕБРАИЧЕСКОЕ ДОКАЗАТЕЛЬСТВО ОСНОВНОЙ ТЕОРЕМЫ АЛГЕБРЫ

The article deals with the presentation of some results obtained in the study of algebraic equations with complex coefficients from a complex variable in a unitary space. In the orthonormal basis, two vectors are introduced, which are called the vector of root and the vector of coefficients of an algebraic polynomial. With the help of these vectors, an algebraic polynomial is represented as a scalar product of them in an orthonormal basis. The criterion of linear independence of a set of root vectors is formulated and proved. A Theorem is formulated and proved that the maximum number of simple roots of an algebraic polynomial is one less than the dimension of a unitary space. A generalization of Vieta’s Theorem is obtained and new formulas connecting the coefficients of an algebraic polynomial with its roots are derived. The general case of an algebraic polynomial is considered, when some of its coefficients may be equal to zero. Two orthogonal vector systems from combinations of coefficients of an algebraic polynomial are introduced. Their properties have been studied. The shortest algebraic proof of the fundamental Theorem of algebra among the known proofs is given, which does not go beyond the concepts of the algebra of polynomials and uses the scalar product of vectors in a unitary space, as well as one property of a two-place predicate from mathematical logic. The results of this work can also be used in the educational process in the algebra course.

Traditional lectures versus active learning – A false dichotomy?

Traditional lectures are commonly understood to be a teacher-centered mode of instruction where the main aim is a provision of explanations by an educator to the students. Recent literature in higher education overwhelmingly depicts this mode of instruction as inferior compared to the desired student-centered models based on active learning techniques. First, using a four-quadrant model of educational environments, we address common confusion related to a conflation of two prevalent dichotomies by focusing on two key dimensions: (1) the extent to which students are prompted to engage actively and (2) the extent to which expert explanations are provided. Second, using a case study, we describe an evolution of tertiary mathematics education, showing how traditional instruction can still play a valuable role, provided it is suitably embedded in a student-centered course design. We support our argument by analyzing the teaching practice and learning environment in a third-year abstract algebra course through the lens of Stanislas Dehaene's theoretical framework for effective teaching and learning. The framework, comprising 'four pillars of learning', is based on a state-of-the-art conception of how learning can be facilitated according to cognitive science, educational psychology and neuroscience findings. In the case study, we illustrate how, over time, the unit design and the teaching approach have evolved into a learning environment that aligns with the four pillars of learning. We conclude that traditional lectures can and do evolve to optimize learning environments and that the erection of the dichotomy 'traditional instruction versus active learning' is no longer relevant.

Evaluación del uso de Wolfram Alpha en el Aprendizaje de Algebra Lineal en Teleducación

The use of ICTs in mathematics learning has the potential to improve academic performance in an interactive environment. Teleducation in COVID times demands the active use of technological tools, thus generating the need to determine which tools are suitable for the different branches of mathematics. This work evaluates the use of the Wolfram Alpha tool in the teaching of Linear Algebra in teleducation in COVID times. This tool was selected because of its ease of use, detailed answers and because it does not require software installation. The research was applied to second semester engineering students, dividing the students into a control group and an experimental group. Pre- and post-treatment test results were statistically analyzed to detect whether there was an increase in academic performance. Students using Wolfram Alpha completed a survey and interview to assess their perception of the tool. The experimental group performed better than the control group. The use of Wolfram Alpha was well accepted by the students suggesting its future use in Linear Algebra courses for tele-education and even face-to-face education in post-pandemic times.

THE DEVELOPMENT OF MATHEMATICAL THINKING THROUGH ADAPTIVE EDUCATION: THE CASE OF A LINEAR ALGEBRA COURSE

The one-size-fits-all model is widely used in teaching and learning processes around the world. This model does not recognize students’ learning conditions and preferences. As an alternative, the adaptive education model has been developing in recent years. In mathematics, the current state of this adaptive education model is oriented to the learning of objects and algorithms and not to the development of mathematical thinking. To overcome this shortcoming, the design-based research approach was used to create and implement an adaptive learning management system framed in a module of a linear algebra course, in which adaptation was done based on modes of thinking (SIERPINSKA, 2000). The results of this implementation show that people manifest diverse modes of thinking when conceiving mathematical objects and that these modes affect the way they learn. These results also highlight some existing limitations for adaptive educational processes such as these to generate greater impacts.

PENGEMBANGAN BUKU AJAR ALJABAR LINIER BERBASIS MATLAB MOBILE UNTUK MENUNJANG PEMBELAJARAN DARING

Learning mathematics online, especially in Linear Algebra courses requires mathematical software-based teaching materials, to strengthen the understanding of numerical concepts. The purpose of this research is to produce MATLAB Mobile-based textbooks that are suitable to support online learning. This development research uses a 4-D model containing 4 stages, namely Define, Design, Develop, and Disseminate. This research was conducted at the Mathematics Education Study Program, the University of PGRI Madiun with 20 students as the subject of the fourth semester. The research instrument and data collection in this study used validation sheets, response questionnaires, and learning outcomes tests. Data analysis of the validation sheet and questionnaire used the Likert scale and the Guttman scale, while the data analysis of the learning outcomes test used the Normalized Gain Score (n-Gain). This study produced a Linear Algebra textbook based on MATLAB Mobile to support online learning during the Covid 19 pandemic with a validity percentage of 86,5% (valid), practicality percentage 83,4% (practical), and an n-Gain score of 46,7 (moderately effectiveness). So that the textbooks developed are suitable to support online learning.