Mathematical Problem Research Articles

Project-Based Learning in Fostering Creative Thinking and Mathematical Problem-Solving Skills: Evidence from Primary Education in Indonesia

The interdependence between the Project-Based Learning (PjBL) Model and the growth and enhancement of Creative Thinking and Mathematical Problem Solving Skills in Elementary Schools is unquestionable nowadays. Prior studies have yet to discover concrete evidence regarding the interdependence being discussed. This study highlighted cognitive abilities related to creative thinking and mathematics problem-solving by implementing the Project-Based Learning Model. This research was a quasi-experiment with a pretest-posttest control group design involving 43 students in the sixth grade of two elementary schools; data was collected through test and classroom observation, and then the data was analyzed using Multivariate Analysis of Variance (MANOVA). Conversely, students exposed to project-based learning models exhibit higher skill levels in creative thinking and problem-solving than those instructed using conventional learning models. The project-based learning model significantly impacted elementary school children’s creative thinking and mathematics problem-solving skills. These findings suggest that the Project-Based Learning Model is acceptable for instructors seeking to foster creativity in teaching mathematics at the primary school level in Indonesia or other countries with comparable settings.

Analysis of students' mathematical communication skills in terms of learning styles

The purpose of this research is to obtain a description of the mathematical communication skills in terms of the learning styles of Grade XI TKR 3 students at SMK Negeri 1 Tanara for the 2023-2024 academic year. This research is qualitative in nature. Six subjects were selected, representing each learning style, from Grade XI TKR 3 students at SMK Negeri 1 Tanara, Tanara District, Serang Regency. The data collection techniques used were observation, interviews, documentation, and tests based on the indicators of mathematical communication skills, namely: (1) the ability to express mathematical symbols through everyday events, (2) the ability to determine the solution set by relating real objects to mathematical ideas, (3) the ability to explain mathematical problems related to elimination, (4) the ability to analyze mathematical problems related to substitution, and (5) the ability to evaluate mathematical thinking and strategies used in problems according to the observed object. The results showed that (1) subjects with a visual learning style have good mathematical concept comprehension on indicators 1, 2, 3, 4, and 5; (2) subjects with an auditory learning style have good mathematical communication on indicators 1, 2, and 5, but performed poorly on indicators 3 and 4; and (3) subjects with a kinesthetic learning style have good mathematical communication skills on indicators 1 and 2, but performed poorly on indicators 3, 4, and 5.

Revealing mental representations of arithmetic word problems through false memories: New insights into semantic congruence.

What can false memories tell us about the structure of mental representations of arithmetic word problems? The semantic congruence model describes the central role of world semantics in the encoding, recoding, and solving of these problems. We propose to use memory tasks to evaluate key predictions of the semantic congruence model regarding the representations constructed when solving arithmetic word problems. We designed isomorphic word problems differing only by the world semantics imbued in their problem statement. Half the problems featured quantities (durations, heights, elevator floors) promoting an ordinal encoding, and the other half used quantities (weights, prices, collections) promoting a cardinal encoding. Across three experiments, in French and in English, we used surprise memory tasks to investigate adults' mental representations when solving the problems. After the first solving task, the participants were given an unexpected task: either to recall the problems (Experiments 1 and 2) or to identify, from memory, the experimenter-induced changes in target problem sentences (Experiment 3). Crucially, all problems included a specific mathematical relationship that was not explicit in the problem statement and that could only be inferred from an ordinal encoding. We used the presence or absence of this relationship in the participants' responses to infer the structure of their representations. Converging results from all three experiments bring new evidence of the role of semantic congruence in arithmetic reasoning, new insights into the relevance of the cardinal-ordinal distinction in numerical cognition, and a new perspective on the use of memory tasks to investigate variations in the representations of mathematical word problems. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Invariant neural architecture for learning term synthesis in instantiation proving

The development of strong CDCL-based propositional (SAT) solvers has greatly advanced several areas of automated reasoning (AR). One of the directions in AR is therefore to make use of SAT solvers in expressive formalisms such as first-order logic, for which large corpora of general mathematical problems exist today. This is possible due to Herbrand's theorem, which allows reduction of first-order problems to propositional problems by instantiation. The core challenge is synthesizing the appropriate instances from the typically infinite Herbrand universe.In this work, we develop a machine learning system targeting this task, addressing its combinatorial and invariance properties. In particular, we develop a GNN2RNN architecture based on a graph neural network (GNN) that learns from problems and their solutions independently of many symmetries and symbol names (addressing the abundance of Skolems), combined with a recurrent neural network (RNN) that proposes for each clause its instantiations. The architecture is then combined with an efficient ground solver and, starting with zero knowledge, iteratively trained on a large corpus of mathematical problems. We show that the system is capable of solving many problems by such educated guessing, finding proofs for 32.12% of the training set. The final trained system solves 19.74% of the unseen test data on its own. We also observe that the trained system finds solutions that the iProver and CVC5 systems did not find.

Pengaruh Aplikasi Matematika Interaktif pada Perkembangan Kemampuan Pemecahan Masalah Matematika

This study aims to determine whether there is an effect of using interactive math applications on the development of mathematical problem solving skills in students. This study was conducted by collecting data from students who used interactive math applications as a tool in learning mathematics, and compared with the control group who used conventional learning methods. Data were collected through math problem solving ability tests before and after the use of interactive math applications. The results showed that the use of interactive math applications significantly improved students' mathematical problem solving ability compared to the control group that used conventional methods. In addition, interactive math applications also help students in understanding math concepts better and increase their interest in this subject.

Analytical model development for vertical medium/deep ground heat exchangers with U-tube(s)

The U-shaped ground heat exchanger (UGHE) offers an effective solution for building heating and cooling purposes. In the practical applications of vertical UGHEs, the borehole wall temperature varies along the borehole depth due to the uneven initial ground temperature and transient heat transfer process. The assumption of a uniform temperature along the borehole wall in the existing quasi-3D model of UGHE causes a non-negligible error in the calculation of fluid temperature, especially for the cases of shallow-medium boreholes. Thus, two new analytical models are established to calculate the fluid temperatures for the single U-shaped GHE (1-UGHE) and double U-shaped GHE (2-UGHE), considering the depth-dependent borehole wall temperatures. The convolution theorem is employed to solve the mathematical problem of the nonlinear condition on the borehole wall, and the Laplace transformation is used for getting the final temperature expressions. The new analytical models are compared and verified by the on-site experiments. The results show that the two models exhibit promising prediction accuracies and similar variation trends in fluid temperatures. Then, parametrical and economical analyses for 1−/2- UGHEs have been carried out. Finally, the geographical thermal potentials of 1−/2- UGHEs across China have been comparatively investigated and the corresponding thermal map-based application potential recommendations have been given.

Students’ Mathematics Performance via APOS Instruction and Learners’ Cognitive Ability on Mathematical Problem Solving: An Analysis

Students’ Mathematics Performance via APOS Instruction and Learners’ Cognitive Ability on Mathematical Problem Solving: An Analysis

TEACHING SURFACE MEASUREMENT UNITS TO ELEMENTARY SCHOOL STUDENTS

Mathematics learning needs to be taught to elementary school children because they will encounter various mathematical problems in their daily lives. The aim of this research is to describe surface material mathematics learning in elementary school children in grades 1 and 2 in Uzbekistan. This research is phenomenological research where researchers describe real situations in the field and report them scientifically. The data collection method used is observation. The results of the research show that teaching surface material mathematics in grades 1 and 2 of elementary schools in Uzbekistan is carried out using the drill/drilling method where the teacher gives students various problems about various types of surfaces. The conclusion obtained from this research is that the drilling method is able to make teachers successful in teaching surface material to students in grades 1 and 2 in elementary schools in Uzbekistan

Basic mathematical skills and fraction understanding predict percentage understanding: Evidence from an intelligent tutoring system

Research on fostering learning about percentages within intelligent tutoring systems (ITSs) is limited. Additionally, there is a lack of data‐driven approaches for improving the design of ITS to facilitate learning about percentages. To address these gaps, we first investigated whether students' understanding of basic mathematical skills (eg, arithmetic, measurement units and geometry) and fractions within an ITS predicts their understanding of percentages. We then applied a psychological network analysis to evaluate interdependencies within the data on 44 subtopics of basic mathematical concepts, fractions and percentages. We leveraged a large‐scale dataset consisting of 2798 students using the ITS bettermarks and working on approximately 4.1 million mathematical problems. We found that advanced arithmetic, measurement units, geometry and fraction understanding significantly predicted percentage understanding. Closer inspection indicated that percentage understanding was best predicted by problems sharing similar features, such as fraction word problems and fraction/natural number multiplication/division problems. Our findings suggest that practitioners and software developers may consider revising specific subtopics which share features with percentage problems for students struggling with percentages. More broadly, our study demonstrates how evaluating interdependencies between subtopics covered within an ITS as a data‐driven approach can provide practical insights for improving the design of ITSs. Practitioner notesWhat is already known about this topic Longitudinal studies showed that basic mathematical skills predict fraction understanding. There is only limited evidence on whether similar predictions can be observed for percentage understanding—in general and within intelligent tutoring systems. Process data from such intelligent tutoring systems can be leveraged to pursue both educational research questions and optimizing digital learning software. Problems involving percentages typically are word problems requiring multiplications and/or divisions. What this paper adds Similar to the case of fractions, students' performance on advanced arithmetic, measurement units and geometry significantly predicted performance with percentages. Students' performance with fractions also predicted performance with percentages significantly. A psychological network analysis was applied to evaluate specific interdependencies between a range of subtopics (eg, Multiplying and dividing fractions, Adding and subtracting fractions and Calculating with percentages). Fraction word problems and fraction problems involving multiplication/division turned out to be the best predictors of understanding percentages. Implications for practice and/or policy When facing difficulties with percentages, revision of previous mathematical concepts sharing similar features (eg, fraction word problems, fraction/natural number multiplication/division problems) may be advised. Software developers may consider implementing such data‐driven revision recommendations for students facing difficulties within intelligent tutor systems. Psychological network analysis can be utilized as a learning analytics method for easy‐to‐access visualizations illustrating relationships between a large range of different subtopics.

Mathematical modelling and optimizing design of heliostat field

The heliostat field is a form of solar thermal power plant, The heliostat field is a crucial component of solar thermal power plants and establishing a heliostat field is of great significance for reducing carbon emissions. and is of great significance in reducing carbon emissions This paper focuses on the mathematical modeling and optimization design problem of the heliostat field. For three different design scenarios, firstly, based on the data provided by the China Undergraduate Mathematical Contest in Modeling 2023, a geometric optical efficiency model is proposed to determine the annual average optical efficiency and output thermal power under specific heliostat field design parameters. Secondly, aiming at the optimal design parameters of the solar furnace field under the condition of rated annual average output thermal power is 60 MW\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext{MW}$$\\end{document}, Particle Swarm Optimization (PSO) combing with efficiency model is utilized to determine the optimal design parameters of the heliostat field. Finally, this paper uses Quasi-Monte Carlo ray tracing (QMCRT) model to perform simulations, exploring how to obtain the optimal design parameters of the heliostat field with the goals of maximizing the output thermal power and ensuring high efficiency. Furthermore, the influence of normal disturbances on the annual average annual average thermal output power is investigated.

Classification of students’ creative thinking for non-routine mathematical problems

Classification of students’ creative thinking for non-routine mathematical problems

Applying a 2D Variational Rigid Block Modeling Method to Rubble Stone Masonry Walls Considering Uncertainty in Material Properties

ABSTRACT The computational cost of detailed micro-modeling of masonry structures can be minimized with rigid block analysis methods, where a variational formulation is particularly interesting as it allows for nonlinear pushover analysis. This work investigates the accuracy of this modeling method in predicting the response of rubble stone masonry walls in shear compression tests. To this end, we extend the mathematical programming problem by including cohesion, compressive strength, and tensile strength in the formulation and implement an improved pushover procedure, which guards compatibility of the displaced shape and crack patterns at the intersection of elastic and rigid contact models. Additionally, we reduce model uncertainty, introduced through sampling material properties with distributions estimated based on results from tests on mortar masonry joints and on mortar specimens, by correlating the predicted crack patterns with the observed ones. The findings show that (i) model falsification based on crack pattern is an effective method for reducing the parameter uncertainty of stone masonry walls; and that (ii) the improved pushover procedure predicts the pre- and post-peak response of stone masonry walls very well with analysis times of less than 20 minutes for 2D models where stones and mortar are modeled explicitly.

Use of mathematical problems rooted in primary historical sources to reveal preservice teachers’ mathematical content knowledge

In this paper, the use of mathematical problems rooted in primary historical sources as a diagnostic tool for identifying learners’ understanding about area and volume is examined. The study follows preservice teachers in the context of a compulsory course in mathematics education while dealing with such problems embedding errors and challenging mistaken beliefs about the area of quadrilaterals and the volume of the cube. Although the initial aim was to involve the participants in inquiry-based activities, in the end, the history of mathematics served as a means to reveal the complete or partial understanding of the concepts making evident at the same time the participants’ misconceptions such as the overgeneralization of the rule ‘length times breadth’ and the confusion between area and perimeter concerning the area of the quadrilaterals task, and the illusion of linearity and the confusion between volume and surface area concerning the volume of the cube task.

The Influence of the Problem Posing Learning Model on Students' Mathematical Problem Solving Abilities in One-Variable Linear Materials

The primary objective of this study is to investigate the impact of the problem posing learning model on the mathematical problem solving abilities of seventh-grade students in the topic of one-variable linear equations. The study aims to determine the extent to which the problem posing learning model influences students' mathematical problem solving abilities at SMPN 2 Gading. Additionally, the study seeks to examine the students' learning activities during the implementation of the problem posing learning model in the context of one-variable linear equations, and to evaluate the effectiveness of implementing the problem posing learning model for seventh-grade students at SMPN 2 Gading. The sampling technique employed in this study is purposive sampling with a simple random sampling approach. The data collection involves essay questions comprising three items to assess students' mathematical problem solving abilities. These questions were pre-tested for validity, reliability, distinguishing power, and level of difficulty. The findings of the research indicate a significant influence of the problem posing learning model on the mathematical problem solving abilities of seventh-grade students at SMPN 2 Gading. The effect size value is reported to be 4.202, indicating a substantial impact. Moreover, the study observes a change in student activities during the learning process, with an increase in student engagement, particularly during the second meeting. The implementation of the problem posing learning model for seventh-grade students at SMPN 2 Gading is reported to have proceeded smoothly from the first to the last meeting.

The Influence Of Self-Confidence On Mathematical Problem Solving Ability Of Senior High School in Indonesia

The Influence Of Self-Confidence On Mathematical Problem Solving Ability Of Senior High School in Indonesia

Use of Language By generative AI Tools in Mathematical Problem Solving: The Case of ChatGPT

Texts generated by artificial intelligence agents have been suggested as tools supporting students’ learning. The present research analyses the language of texts generated by ChatGPT when solving mathematical problems related to the quadratic equation. We use the functional grammar theoretical framework that includes three meta-functions: the ideational meta-function, the interpersonal meta-function and the textual meta-function. The results indicated that in at least one of six problem-solving tasks ChatGPT provided a mathematically incorrect answer. The processes appearing in ChatGPT texts, aiming at developing students’ understanding of mathematical concepts, included verbal, mental, existential, relational and behavioural processes but no material processes. Specifically, ChatGPT performed a mathematically incorrect existential process. ChatGPT generally used the first plural pronoun ‘we’ when describing the processes of solving mathematical problems, while it generally used the first-person singular pronoun when taking responsibility for a specific mistake or when expressing happiness for the actions of the user. Moreover, generally the text of the solution did not include direct imperatives but used ‘let us do’. The advancement of the ChatGPT textual solution was made usually through steps like ‘first’, ‘second’, etc. The research results indicated that the way ChatGPT responded to the mathematical problems would be useful in supporting learners’ understanding of ways to solve quadratic equations, but only if the teacher critically accompanies the student in the problem-solving process. Self-study with ChatGPT could lead to or confirm students’ mathematical misconceptions.

Systematic Review in Solving Mathematical Problems

The ongoing challenges faced by mathematics students are highlighted through an analysis of the effectiveness of different teaching styles used worldwide. A systematic review of the literature selected nine articles from a total of 598 relevant articles using rigorous inclusion criteria and quantitative techniques. The results suggest that incorporating problem-solving skills into educational programs can significantly improve students' critical thinking and academic performance. Constructivist and sociocultural approaches, which promote collaboration, introspective practice, and participation, have been shown to be highly effective. The conclusion is that to foster a fruitful and meaningful learning environment and to enhance students' critical thinking and academic achievement in mathematics, a balanced teaching approach that combines direct instruction and independent learning is essential. This can be achieved through the use of technology and collaborative resources.

Examining the role played by resources, goals and orientations in primary teachers’ decision- making for problem-solving lesson plans

The decisions that teachers make in transforming the curriculum into specific lesson plans determine the real enactment or otherwise of curricular ideals. These decisions are shaped by the resources available and by each teacher’s goals and orientations. This exploratory study employs Schoenfeld’s decision-making model to examine how resources, goals and orientations influence lesson planning for mathematics problem solving, for different profiles of primary teachers in Chile. To this purpose, a survey was conducted among 40 teachers of varying degrees of ability and experience: some were beginning teachers, others were experienced but had no further training in teaching problem solving and a third group was composed of experienced teachers with specific training in this question. Interviews with two teachers from each profile revealed important differences between the three groups. Beginning teachers relied more heavily on official resources such as the official curriculum and standard textbooks, aligning themselves with school requirements. Experienced teachers with problem solving training demonstrated a strong inclination towards teaching through a problem solving approach. While beginning teachers acknowledged the importance of promoting problem solving strategies, they did not usually adapt problems to the mathematical content or to the age/competence of their students. Interestingly, all three groups under-utilised sections of curricular resources that emphasise the present curricular focus on problem solving. Finally, the study found that experience alone is not enough to develop a problem solving approach and that focused professional development programmes are needed to equip teachers with the necessary skills. In addition, a diagnostic teaching approach should be incorporated into initial teacher training.

Representation and Generation of Prime and Coprime Numbers by Using Structured Algebraic Sums

The algebraic structure and distribution of prime numbers remain two of the most fundamental problems in mathematics. The Fundamental Theorem of Arithmetic, proved by Euclid, and Goldbach’s conjecture, while universal in scope with respect to how numbers can be represented multiplicatively or additively, do not provide insights into the structure of primes. Similarly, the definition of a prime −as a number divisible only by 1 and itself− or a sieve algorithm, commonly used to generate primes by successively eliminating multiples, offer no insight into the structure of primes. The powerful and persistent consideration of prime numbers as universal “arithmetic quanta” has not necessitated an equally powerful need for parallel research into a deeper and possibly more insightful explanation of primeness, that is, a better understanding of “why” a number is prime. In this paper, prime and coprime numbers are represented and generated by algebraic expressions. Specifically, given the first <i>n</i> primes, <i>p<sub>1</sub></i>,<i> p<sub>2</sub></i>,…, <i>p<sub>n</sub></i>, sufficient conditions are given for expressing primes greater than <i>p<sub>n</sub></i>, and coprimes with prime factors greater than <i>p<sub>n</sub></i>, as algebraic functions of <i>p<sub>1</sub></i>,<i> p<sub>2</sub></i>,…, <i>p<sub>n</sub></i>. Thus, primality and co-primality are shown to be mathematical properties with inherently evolutionary algebraic characteristics, since larger primes and coprimes can be generated algebraically from smaller ones. The methodology described in the paper can be a useful tool in the study and analysis of the complexity, structure, interrelationships and distribution of primes and coprimes.

Mathematical Concept Representations in the Early Years of School: Analyzing Student’s Errors

In this paper we analyze students’ mathematics errors in the early years of Brazilian school. To achieve this goal, we developed a questionnaire with eight math problems that included basic operations. This instrument was distributed to 76 third-grade students in three different classes from three different schools. To comprehend what caused the errors, we analyzed each question. To perform these analyzes we used the following categories: Errors due to language difficulties; Errors due to difficulties in obtaining spatial information; Errors due to incorrect associations or rigidity of thinking; Errors due to the application of irrelevant rules or strategies; Errors due to deficient mastery of prerequisite skills, facts, and concepts; Incorrect reproduction of the task; Error of counting; Errors in the assemblage of the arithmetic operations; Errors in the summing up with values bigger than ten; and Specific errors of subtraction. We believe that using errors as motivation could improve students' understanding of mathematics. We seek to help teachers (and researchers) identify most of the mistakes made by students in order to recognize them in their classrooms. The results indicated that some of the students' most common errors are in line with what the literature points out, that is, difficulty in translating the problem into mathematical language; difficulties in using the operations algorithms and difficulties in understanding what the problem is asking.