Enhancing Understanding of Linear Programming Concepts Through Contextual Education

This essay investigates the concept of linear programming in general and linear stochastic programming in particular. Linear stochastic programming is described as the model where the parameters of the linear programming admit random variability. The first three chapters present through a set-geometric approach the foundations of linear programming. Chapter one describes the evolution of the concepts which resulted in the adoption of the model. Chapter two describes the constructs in n-dimensional euclidian space which constitute the mathematical basis of linear programs, and chapter three defines the linear programming model and develops the computational basis of the simplex algorithm. The second three chapters analyze the effect of the introduction of risk into the linear programming model. The different approaches of estimating and measuring risk are studied and the difficulties arising in formulating the stochastic problem and deriving the equivalent deterministic problems are treated from the theoretical and practical point of view. Multiple examples are given throughout the essay for clarification of the salient points.

This study was motivated by limited comprehension of mathematical concepts among students in mathematics. One potential solution is implementing the small group discussion learning method. The objective of this study is to ascertain the efficacy of the acquisition of small-group discussion methods in mathematics subjects on students' comprehension of mathematical concepts. This form of research is a quasi-experiment with a non-equivalent control group design. The samples in this study were class , which consisted of 27 students as the experimental class, and , which consisted of 28 students as the control class, selected using Simple Random Sampling. The t-test results indicated a substantial difference in the comprehension of mathematical concepts between the experimental and control class students. Consequently, H0 was rejected, and H1 was accepted. It can be concluded that the small group discussion method is more effective than conventional learning in enhancing students' comprehension of mathematical concepts. The small group discussion method effectively enhances students' comprehension of concepts by involving them directly in the learning process. The small group discussion method has the potential to enhance students' comprehension of concepts, as demonstrated by the empirical findings of this study. Thus, this study suggests that this method is an effective teacher learning strategy.

The research investigates the effect of remedial practices on students’ competencies in mathematics subjects in Rwanda, in Kamonyi district. Specifically, the research identified remedial practices from public secondary schools that affect students’ competencies in mathematics in Kamonyi district, Rwanda, and analyzed the students' performance in mathematics that is due to remedial practices in secondary schools in Kamonyi district and determined the relationship between remedial practices and students’ competencies in mathematics in Kamonyi district, Rwanda. The population in this study consisted of 5331 respondents, while their sample size was 370. To triangulate the data, primary sources were acquired utilizing questionnaires, interviews, and observation methods. To generate a sample population from the respondents, this study employed purposive, stratified, and simple random sampling methods. In data gathering and analysis, the study used both quantitative and qualitative methodologies in tandem. Content analysis aided qualitative data analysis, while quantitative data was presented using descriptive statistics (frequency, percentage, mean, and standard deviation) and inferential statistics (correlational and regression analysis) in IBM SPSS Version 21.0. For the first objective, results indicate that 100.0% of respondents either strongly agreed or agreed that changing teaching methods frequently, 90.0% of respondents either strongly agreed or agreed that using individualized learning materials helps in remedial practices in mathematics, 90.0% of respondents either strongly agreed or agreed that incorporating visual and audio resources, and 80.0% either strongly agreed or agreed that creating original worksheets and exercises as remedial practices in mathematics. For the second objective, 60.0% of respondents 100% either strongly agreed or agreed that the student’s problem-solving ability, 80.0% of respondents either strongly agreed or agreed that the student’s ability to reason, 90.0% of respondents either strongly agreed or agreed that the ability to apply methods by students demonstrates, and 80.0% of respondents either strongly agreed or agreed that the capacity for representing mathematical structures indicates the student's competencies in mathematics. The study found a strong relationship between creating lessons that consider students' interests, needs, and experiences and their problem-solving abilities, procedure application abilities, and capacity for representation. It also found a positive relationship between the use of individualized learning materials and problem-solving abilities, procedure application abilities, and capacity for representation. The results suggest that adjusting these factors can enhance problem-solving, procedure application, and representation capacity. These findings suggest that they are correlated since most of their levels of significance were greater than 0.05 in association with students’ competencies in mathematics subjects in Rwanda, in Kamonyi district. It is recommended that Rwandan Education Board and school: REB, as well as the school level, are recommended schools to use remedial activities, and the school level applies this program to facilitate learners to increase their achievement and encourage their learning. To teachers of mathematics: Teachers should always make sure that students feel comfortable with the method they are using in teaching mathematics to facilitate learners to solve daily problems. Teachers should mobilize their students and teachers to attend remedial activities. Teachers give more exercises, practical activities, and homework. Teachers encourage students to learn and like mathematics subjects as an important subject in daily life.

The use of linear programming in decision making has been widespread for twenty to thirty years, with its theoretical and solution concepts well documented and explored. In spite of its wide use one main difficulty in linear programming remains, and that is the fact that linear programming problems are defined with only one objective function. A. Charnes and W. W. Cooper, through continuing research, developed the concept of goal programming which enables one to solve linear programming problems with multiple objectives. This dissertation presents computational procedures which will enable one to solve a large-scale linear goal programming problem as well as its multidimensional dual problem. Applications in statistics and industry are explored.

Chapter 7 - Linear Programming

Problem solving is one of the competencies in learning mathematics. It is expected that students have this competency to be able to solve problems in their life by using math. The aim of this research is to improve problem solving competence through contextual learning based on coastal culture. The research design used was pretest-posttest control group design. The population is junior high school students in East Flores District, East Nusa Tenggara Province. 119 students were chosen by random purposive sampling technique. Four classes from grade 8 from a Junior High School at Larantuka were randomly selected as experimental and control class. The experimental class students employ contextual learning based on coastal culture, whereas students in the control class use conventional learning. The instrument of this research is test of mathematical problem solving competence. The results showed that there was an enhancement in mathematical problem solving competence in students who received contextual learning based on coastal culture (CLBCC) and conventional learning approach (CLA). Improvement competence of mathematical problem solving on CLBCC group is higher than CLA group. This also happened to the achievement of mathematical problem-solving competency. There are differences in the improvement and achievement of mathematical problem-solving competency between CLCBB and CLA students.

SimplifEx: Simplifying and Explaining Linear Programs

The assessment of mathematical competence, particularly in real-world problem-solving contexts, has become increasingly crucial in high school educational evaluation. While traditional methods have shifted towards emphasizing problem-solving skills, they remain predominantly outcome-oriented, often failing to adequately capture the nuanced cognitive processes underlying students’ problem-solving behaviors. To address this gap, this study introduces a process-oriented assessment method leveraging eye-tracking technology. Fifty-three university students (primarily first- and second-year undergraduates) in China were recruited to solve six context-based mathematical problems of varying difficulty levels while wearing portable eye-tracking glasses, allowing for natural problem-solving behaviors in a paper-and-pencil test format. The study established a multidimensional model of eye movement features to evaluate problem-solving processes. Using China’s National College Entrance Examination (CNCEE) mathematics scores as the dependent variable, a Partial Least Squares Regression (PLSR) analysis achieved its best predictive performance (prediction R2 of 0.271) based on multidimensional eye movement features when solving the most difficult problem. The first visual intake duration on problem-reading areas and key information regions emerged as significant contributors of the students’ CNCEE scores. These findings substantiate the potential of eye-tracking technology as a valuable tool for educational assessment, offering insights into the assessment of students’ mathematical competence and supporting the development of more comprehensive learning diagnosis and intervention strategies.

This paper aims to investigate the impact of library on learners; linear programming performance, employing a qualitative approach. In this study, the effect of libraries on students’ performance in linear programming is examined using a qualitative methodology. The author claims that because libraries give students access to study materials from the actual school library, they can aid in their understanding of linear programming. The author argues that effective learning of linear programming requires studying. Libraries offer a surfeit of services and resources that assist academic accomplishment, making them useful tools for student’s at all educational levels. Libraries help students with complex topics and assignments by providing specialist research assistance and reference services in addition to textbooks, reference materials, and scholarly journals. Building school libraries is one way to encourage learners to put in a lot of effort in their study work, according to the study’s findings. This article’s goal is to disseminate the results of a qualitative study that highlights the benefits of the school library. This has the potential to improve education and inspire students to work even harder in their academic endeavors. Additionally, students who have difficulty in class can borrow books from the school library. The library will support students’ positive attitudes toward learning and active engagement in class activities. To reduce poor performance, school should ensure that library is put in place. This will motivate learners to put extra effort into their studies. The study concludes by recommending the establishment of building more schools libraries and stocking them relevant study materials. Parents can also ask school administrators to guide them to buy the required study materials for their children to study.

In this book the author analyzes and compares four closely related problems, namely linear programming, integer programming, linear integration, linear summation (or counting). The focus is on duality and the approach is rather novel as it puts integer programming in perspective with three associated problems, and permits one to define discrete analogues of well-known continuous duality concepts, and the rationale behind them. Also, the approach highlights the difference between the discrete and continuous cases. Central in the analysis are the continuous and discrete Brion and Vergne's formulae for linear integration and counting. This approach provides some new insights on duality concepts for integer programs, and also permits to retrieve and shed new light on some well-known results. For instance, Gomory relaxations and the abstract superadditive dual of integer programs are re-interpreted in this algebraic approach. This book will serve graduate students and researchers in applied mathematics, optimization, operations research and computer science. Due to the substantial practical importance of some presented problems, researchers in other areas will also find this book useful.

This study seeks reflection on the approaches of 11th grade students to Linear Programming problems, discussing the approaches taken at different moments of the teaching process. It aims to analyze: How is the students’ mathematical competence characterized in relation to problem- solving; What differences can be identified in the resolutions at different moments of the teaching and learning process. We adopt a qualitative and interpretative methodology, analyzing the approaches of two pairs of students with different mathematical backgrounds. The analysis is guided by Pólya's stages of solving a problem and aspects of the understanding of mathematical competence. The results show different approaches to the problems depending on the teaching moment and different competences. The mathematical background impacts the students’ success when they implement routine procedures, however it does not seem to determine the students’ competence to reason about a problem.

The present data was applied to investigate the relationship between students’ attitude towards, and performance in mathematics word problems (MWTs), mediated by the active learning heuristic problem-solving (ALHPS) approach. Specifically, the data reports on the correlation between students' performance and their attitude towards linear programming (LP) word tasks (ATLPWTs). Four types of data were collected from 608 grade 11 students who were selected from eight secondary schools (both public and private). The participants were from two districts: Mukono and Mbale in Central Uganda and Eastern Uganda respectively. A mixed methods approach with a quasi-experimental non-equivalent group design was adopted. The data collection tools included standardized LP achievement tests (LPATs) for pre-test and post-test, the attitude towards mathematics inventory–short form (ATMI-SF), a standardized active learning heuristic problem-solving tool, and an observation scale. The data were collected from October 2020 to February 2021. All the four tools were validated by mathematics experts, pilot-tested, and found reliable and suitable for measuring students’ performance and attitude towards LP word tasks. To achieve the purpose of the study, eight intact classes from the sampled schools were selected using the cluster random sampling method. Of these, and by a toss of a coin, four were randomly assigned to the comparison group while the other four were randomly assigned to the treatment group. All teachers from the treatment group were trained on the application of the ALHPS approach before intervention. The participants’ demographic data (identification numbers, age, gender, school status, and school location) were presented together with the pre-test and the post-test raw scores obtained before and after intervention respectively. The LPMWPs test items were administered to the students to explore and assess their problem-solving (PS), graphing (G), and Newman error analysis strategies. Students' percentage scores in the pre-test and post-test were based on mathematizing word problems to optimization of LP problems. Data were analyzed in line with the purpose of the study, and the stated objectives. This data supplements other data sets and empirical findings on the mathematization of mathematics word problems, problem-solving strategies, graphing and error analysis prompts. This data may serve and provide some insights into the extent to which ALHPS strategies support students' conceptual understanding, procedural fluency, and reasoning among learners in secondary schools and beyond. The LPMWPs test items in the supplementary data files can also act as a basis for the application of mathematics in real-life scenarios beyond the compulsory level. The data is intended to develop, support, and strengthen students' problem-solving and critical thinking skills with the main goal of enhancing instruction and assessment in secondary schools and beyond.

The purpose of this study was to improve students' mathematical literacy competence in class XI at SMA YPK Medan by implementing the Realistic Mathematics Education (RME) approach supported by educational technology. Mathematical literacy includes students' ability to understand, interpret, and apply mathematical concepts in real-world contexts. In this study, an experimental approach with pre-tests and post-tests was used to compare the results of the control and experimental classes. The experimental class was taught using the RME approach supported by technology such as mathematics learning applications and interactive media, while the control class used traditional teaching methods without technological support. Data were analyzed using a t-test to analyze differences in mathematical competence between the two classes. The results of this study indicate that the RME approach supported by technology is significantly more effective in improving students' mathematical literacy compared to traditional methods, especially in terms of formulating, interpreting, and applying mathematics to solve real-world problems. This research also shows that the integration of technology in mathematics learning can enrich students' learning experiences, increase engagement, and support the development of better mathematical skills. Abstrak Tujuan dari penelitian ini adalah untuk meningkatkan kompetensi literasi matematika siswa pada kelas XI di SMA YPK Medan dengan menerapkan Pendekatan Matematika Realistik (PMR) yang didukung oleh teknologi pendidikan. Literasi matematika mencakup kemampuan siswa dalam memahami, menginterpretasikan, dan mengaplikasikan konsep-konsep matematika dalam konteks dunia nyata. Dalam penelitian ini, pendekatan eksperimental dengan pre-test dan post-test digunakan untuk membandingkan hasil dari kelas kontrol dan kelas eksperimen. Kelas eksperimen diajar menggunakan pendekatan PMR yang didukung oleh teknologi seperti aplikasi pembelajaran matematika dan media interaktif, sedangkan kelas kontrol menggunakan metode pengajaran tradisional tanpa dukungan teknologi. Data dianalisis dengan menggunakan uji-t untuk menganalisis perbedaan kompetensi matematika antara kedua kelas. Hasil penelitian ini menunjukkan bahwa pendekatan PMR yang didukung oleh teknologi secara signifikan lebih efektif dalam meningkatkan literasi matematika siswa dibandingkan dengan metode tradisional, terutama dalam hal merumuskan, menafsirkan, dan mengaplikasikan matematika untuk memecahkan masalah dunia nyata. Penelitian ini juga menunjukkan bahwa integrasi teknologi dalam pembelajaran matematika dapat memperkaya pengalaman belajar siswa, meningkatkan keterlibatan, dan mendukung pengembangan keterampilan matematika yang lebih baik.

Acquiring mathematical literacy requires students to apply mathematics in various real-world contexts. However, mathematics classes often provide brief, content-focused descriptions of reality-based tasks and tasks that describe the situation as more complex, closer to reality, are still lacking. Students with different sociodemographic characteristics and cognitive factors have difficulties in solving reality-based tasks in mathematics lessons. The relationship between sociodemographic characteristics and cognitive factors (language and mathematical competence) concerning complex situation descriptions has not yet been investigated. To identify disadvantaged students in integrating such complex-situation tasks in mathematics lessons, this study aims to investigate which sociodemographic characteristics predict the solving of complex-situation tasks and whether cognitive factors mediate the relationship. Experts created 30 complex situations with different mathematical questions. A total of 519 9th- and 10th-grade students participated in a paper–pencil test. Path analysis revealed that the competence to solve complex-situation tasks is directly linked to gender and social background, with mathematical content-related skills and language competence mediating this relationship.

A number of national and international scale surveys showed that the mathematics achievements of Indonesian secondary school students were in the lower level. One of the indicators of this low achievement was the insufficient level of understanding of mathematical concepts of the students. Understanding of mathematical concepts could be integrated through Realistic Mathematics Education (RME). Therefore, the quasi-experimental study was conducted to examine the effectiveness of Realistic Mathematical Education towards the conceptual understanding of linear programming. The study also investigated the relationship between conceptual understanding and mathematics achievement and investigated the misconceptions on linear programming. This study involved 65 students of Madrasah Aliyah Negeri 1 Pekanbaru, Indonesia. Test conceptual understanding programming topics had Cronbach’s alpha reliability of 0.80. The data were collected by using a test question conceptual understanding of linear programming. Quantitative data analysis involved a descriptive and inferential analysis using SPSS 21.0. The descriptive analysis included the percentage, mean and standard deviation while inferential analysis involving independent t-test and Pearson correlation analysis. The results revealed that there were significant differences between the treatment and control groups toward conceptual understanding. There was a significant relationship between conceptual understanding and mathematics achievement of linear programming. Misconception toward linear programming for the treatment group was lower than the control group. The implications of this study were useful for educators to help their students to understand concept of mathematics through open and contextual questions so that the students though through the mathematics, not worked in the mathematics.