Basic Arithmetic Research Articles

Peningkatan Kemampuan Operasi Hitung Siswa Kelas VII Melalui Program Kelas Numerasi SMPN 16 Mataram

Numeracy skills are fundamental abilities that every student must possess to understand mathematical concepts. Indonesia ranks 74th out of 79 countries in terms of numeracy proficiency, indicating the low ability of students to comprehend basic mathematical concepts and solve numeracy-based problems. According to the 2024 National Assessment results, the majority of students in this region fall into the low category for numeracy literacy. This is further supported by data from SMPN 16 Mataram, where field observations reveal that many students struggle with mastering basic arithmetic operations such as addition, subtraction, multiplication, and division, which are foundational for understanding more advanced mathematical concepts. Therefore, this community service aims to help students develop skills in understanding and operating basic algebraic concepts, such as addition, subtraction, multiplication, and division of algebraic expressions. This service is conducted at SMPN 16 Mataram, located in Mataram city, with the participants being 23 students from class VII of SMP in Mataram. The data used in this service activity is sourced from the implementation of the Madrasah Indonesia Competency Assessment (AKM) guidance activity to improve students' literacy and numeracy competencies at SMPN 16 Mataram, conducted by 7th grade mathematics teacher. The evaluation of the activity was done through interviews with both the teacher and the students of grade 7th. The results of this activity show a significant improvement in the numeracy skills of students in algebraic arithmetic operations at SMPN 16 Mataram, class VII. This improvement is measurable, with the average score initially at 40, which then increased to 70 after the implementation of the program or community service activities. This indicates that students who previously had difficulty understanding and mastering algebraic arithmetic operations, such as addition, subtraction, multiplication, and division with variables or algebraic expressions, made significant progress after being given guidance and practice. Therefore, the addition of numeracy-focused programs is effective in helping students solve problems.

Integrating Transformer Architecture and Householder Transformations for Enhanced Temporal Knowledge Graph Embedding in DuaTHP

The rapid advancement of knowledge graph (KG) technology has led to the emergence of temporal knowledge graphs (TKGs), which represent dynamic relationships over time. Temporal knowledge graph embedding (TKGE) techniques are commonly employed for link prediction and knowledge graph completion, among other tasks. However, existing TKGE models mainly rely on basic arithmetic operations, such as addition, subtraction, and multiplication, which limits their capacity to capture complex, non-linear relationships between entities. Moreover, many neural network-based TKGE models focus on static entities and relationships, overlooking the temporal dynamics of entity neighborhoods and their potential for encoding relational patterns, which can result in significant semantic loss. To address these limitations, we propose DuaTHP, a novel model that integrates Transformer blocks with Householder projections in the dual quaternion space. DuaTHP utilizes Householder projections to map head-to-tail entity relations, effectively capturing key relational patterns. The model incorporates two Transformer blocks: the entity Transformer, which models entity–relationship interactions, and the context Transformer, which aggregates relational and temporal information. Additionally, we introduce a time-restricted neighbor selector, which focuses on neighbors interacting within a specific time frame to enhance domain-specific analysis. Experimental results demonstrate that DuaTHP significantly outperforms existing methods in link prediction and knowledge graph completion, effectively addressing both semantic loss and time-related issues in TKGs.

"Groupitizing": A Visuo-Spatial and Arithmetic Phenomenon.

When objects are grouped in space, humans can estimate numerosity more precisely than when they are randomly scattered. This phenomenon, called groupitizing, is thought to arise from the interplay of two components: the subitizing system which identifies both the number of subgroups and of items within each group, and the possibility to perform basic arithmetic operations on the subitized groups. Here we directly investigate the relative role of these two components in groupitizing via an interference (dual task) paradigm. Participants were required to estimate numerosities of grouped and ungrouped arrays while their attentional resources were fully available (single task) or while performing concurrent tasks loading auditory or visuo-spatial attention (both known to interfere with the subitizing process) as well as while performing arithmetic calculation. The attentional cost of performing any concurrent task was overall higher for grouped than ungrouped stimuli, supporting the idea that groupitizing relies on the recruitment of more than one attention-dependent mechanism. However, depriving visuo-spatial attention and preventing participants from performing calculations caused the strongest decrement in sensory precision for grouped numerosities indicating that these attentional components play a major role in groupitizing. These results are in line with the existence of an estimation mechanism that might operate across all numerical ranges, supplemented by attentional mechanisms (subitizing). This study shows that this attentional-demanding mechanism can be activated also when processing numerosities outside of the subitizing regime (n > 4), provided that grouping cues are available and, in concert with calculation abilities, gives rise to the groupitizing phenomenon.

The calculating brain.

The human brain possesses neural networks and mechanisms enabling the representation of numbers, basic arithmetic operations, and mathematical reasoning. Without the ability to represent numerical quantity and perform calculations, our scientifically and technically advanced culture would not exist. However, the origins of numerical abilities are grounded in an intuitive understanding of quantity deeply rooted in biology. Nevertheless, more advanced symbolic arithmetic skills require a cultural background with formal mathematical education. In the past two decades, cognitive neuroscience has seen significant progress in understanding the workings of the calculating brain through various methods and model systems. This review begins by exploring the mental and neuronal representations of nonsymbolic numerical quantity and then progresses to symbolic representations acquired in childhood. During arithmetic operations (addition, subtraction, multiplication, and division), these representations are processed and transformed according to arithmetic rules and principles, leveraging different mental strategies and types of arithmetic knowledge that can be dissociated in the brain. Although it was once believed that number processing and calculation originated from the language faculty, it is now evident that mathematical and linguistic abilities are primarily processed independently in the brain. Understanding how the healthy brain processes numerical information is crucial for gaining insights into debilitating numerical disorders, including acquired conditions like acalculia and learning-related calculation disorders such as developmental dyscalculia.

School Teachers’ Perspectives on the Use of Algebra Tiles to Operations on Integers

This paper is based on the first author’s (researcher) experience as a mathematics teacher trainer based on his M.Ed. ‘Internship course.’ The second author being the course facilitator has played a facilitative role in the development of the training module and the successful conduction of Teacher Professional Development (TPD) sessions. This study aimed to explore perspectives on the use of algebra tiles (as physical manipulatives) to basic arithmetic operations on integers - addition ( ), subtraction ( ), multiplication ( ), and division ( ), based on the experiences of 22 school mathematics teachers of Kathmandu, Nepal (who participated in TPD). This study utilized a qualitative research technique, Concrete-Representational-Abstract (C-R-A) instructional sequence, and triangulation. The data generation tools were participant observations, researcher-participant interaction, participant worksheets, reflection forms, end-reflection videos, and photos. The data analysis unveiled that the participants had a positive outlook towards the use of algebra tiles (as physical manipulatives), and they viewed it as effective concerning operations on integers. As a result, the participants were motivated enough and were determined to apply it in their mathematics classroom teaching, even though it would be difficult to implement it properly due to time constraints – cost, availability, time, and management. However, the participants suggested Book Free Day - an initiative introduced (2023, April 27) by Kathmandu Metropolitan City, where students go to school without books and bags every Friday, and remedial teaching as the most appropriate occasions to implement algebra tiles properly.

Description of Process Skills of Students with Dyscalculia in Understanding the Concept of Counting Operations Through Ethnomathematics

Mathematics is fundamental in daily life and education, yet students with dyscalculia face significant challenges in mastering basic arithmetic operations. This study investigates how the ethnomathematics approach can develop process skills among students with dyscalculia. Using a qualitative descriptive method, two grade-six students with different levels of dyscalculia were studied through observations, tests, and interviews. Results indicate that students with moderate dyscalculia (S1) demonstrated progress in classification and calculation with minimal guidance, while those with severe dyscalculia (S2) required intensive support and showed limited progress. This suggests that ethnomathematics, which integrates cultural and contextual learning, is effective for students with learning disabilities but requires adaptation based on severity levels. Findings highlight the potential of culturally relevant methods in enhancing mathematical skills.

Effectiveness of Basic Arithmetic Skills Module among Pupils with Dyscalculia Symptoms

Mathematics is fundamental in the educational system, yet many pupils lack the required skills in the subject. Intervention is crucial for pupils with dyscalculia symptoms. This study evaluates the effectiveness of the Basic Arithmetic Skills (BAS) Module among pupils with dyscalculia symptoms. This is a quantitative study with a quasi-experimental design. One hundred twenty pupils aged between 7 and 14 years, who exhibited dyscalculia symptoms were involved in this study. They were divided into two (2) groups: the control group with 59 pupils and the treatment group with 61 pupils. The results showed that the BAS Module significantly improved the basic arithmetic skills among pupils with dyscalculia symptoms regardless of gender and age. In conclusion, the BAS Module effectively improved the basic arithmetic skills among pupils with dyscalculia symptoms in primary schools. The study emphasizes the importance of tailored instructional materials and the role of teachers in delivering interventions that cater to pupils' different cognitive development levels. It is recommended that the BAS Module be applied to pupils with dyscalculia to improve their basic arithmetic skills, particularly in addition, subtraction, multiplication, and division.

Gamification Practices and Student Academic Performance in Mathematics in Public Secondary Schools in Rwanda

This research examined the impact of gamification techniques on math proficiency among students in Rwandan public secondary schools. In particular, to look at how gamification techniques affect math performance in Rwandan public secondary schools, to gauge how much math performance is attributable to gamification techniques in Rwandan public secondary schools, and to analyse the connection between gamification techniques and math academic achievement in Rwandan public secondary schools. 503 pupils, 52 headteachers, and 115 instructors made up the study's population ; there were 670 responders in all, and 251 kids from 20 schools in the Muhanga District made up the sample size. Results were analysed using SPSS version 21. According to the results for the first objective, 57.4% strongly agreed and 19.1 % agreed that assigning points, badges, or other rewards for completing math problems or achieving specific milestones can encourage them to engage more deeply with the material. 58.5% strongly agreed and 18.1% agreed that displaying a leaderboard that shows my progress and rankings compared to my peers can foster healthy competition and motivate them to improve. 68.1% strongly agreed that introducing virtual currencies or tokens that students can earn for completing math tasks can add an element of fun and encourage students to participate more actively. 66.0% strongly agreed and 21.3% agreed that designing math challenges or quests that students can complete to earn rewards or advance through a game-like progression system can make learning math more engaging and enjoyable and 74.5% strongly agreed and 21.3% that using interactive simulations or math games that allow students to visualise and interact with mathematical concepts can help students better understand and retain mathematical concepts. To the second aim, 53.5% strongly agreed and 16.3 % agreed that their ability to accurately solve math problems and complete assignments can indicate their understanding of mathematical concepts and my ability to apply them to real-world scenarios, while 53.5% agreed and 20.9% that their fluency and speed in performing math operations, such as addition, subtraction, multiplication, and division, can indicate their comfort and proficiency with basic arithmetic. 65.1% strongly agreed and 18.6% that their ability to solve complex math problems and demonstrate critical thinking skills can indicate their understanding of mathematical concepts and their ability to apply them to novel situations, and finally, 60.5% strongly agreed and 18.6% Agreed that their ability to explain and demonstrate their understanding of mathematical concepts, such as fractions, decimals, or algebra, can indicate their grasp of the subject matter. The correlation matrix between independent variables (problem-based learning, displaying leaderboards, using projectors in teaching math, and assigning badges instead of grades) and dependent variables (improved student mathematics score from tests and exams, exam and test results, engagements and participation, and improved class participation) Finally, they indicate positive significance since the p-value is smaller than 0.05. The study recommended that Incorporating game-like elements, interactive tools, and collaboration into the math curriculum can make learning more engaging and motivate students. These tools can help students understand complex concepts in a fun and engaging way. Providing immediate feedback on performance, setting clear goals, and creating a positive learning environment can also help students stay focused and motivated. These gamification practices can lead to improved academic performance in math in public secondary schools in Rwanda.

Design and Implementation of a 3-bit ALU with Integrated 7-Segment Display on FPGA

This paper describes the design and implementation of a 3-bit Arithmetic Logic Unit (ALU) with integrated 7-segment display output, deployed on an FPGA platform. The objective of this project is to efficiently perform and display basic arithmetic and logic operations, including addition, subtraction, AND, OR, XOR, and NOT, with an FPGA-based ALU. A control signal selects the operation, and the ALU's 3-bit result is decoded and displayed on a 7-segment interface for clear output visualization. The ALU design is coded in Verilog and includes logic to manage carry and overflow in arithmetic functions. A display decoder module maps the ALU’s binary output to the correct LED segments on the 7-segment display, allowing intuitive real-time readout of operation results. The system was synthesized, simulated, and tested on an Artix-7 FPGA using the Xilinx Vivado development environment, achieving accurate and expected functionality across all operations. This work showcases the effectiveness of FPGA technology in integrating computation and display within a compact digital system, with potential applications in both educational and embedded system design settings where real-time processing and display are essential. Keywords: FPGA, ALU, 7-segment display, Verilog, Digital system design, Embedded computing.

Asymmetric interval numbers: A new approach to modeling uncertainty

This paper introduces Asymmetric Interval Numbers (AINs), a novel type of interval numbers that combines the straightforward identification characteristic of classical interval numbers with the advanced capability for modeling uncertainty found in fuzzy sets, all while maintaining simplicity. AINs incorporate the expected value within the interval, providing a more accurate representation of the uncertainty of the data compared to the classical interval numbers. We define basic arithmetic operations for AINs, discuss their properties, and provide proofs. Additionally, we present theorems on symmetry and asymmetry for fundamental binary and unary operations, enhancing the mathematical framework for AINs. Several illustrative examples demonstrate the practical application of AINs. Although challenges remain, such as exploring performance with different distributions and reducing overestimation, AINs present a promising advancement in interval arithmetic. This study underscores the practical and theoretical implications of AINs, paving the way for further research and application in diverse scientific and industrial contexts.

Introducing variables in the evolution rules of P systems

In membrane systems evolution rules are constructed using only objects from a finite alphabet. In this paper we investigate rules in which variables are used. Namely, we define Variable P systems in which the rules containing variables need to be instantiated at the start of each computational step with values from some predefined sets of sets of objects.The modelling power of variable P systems is described by simulating some basic arithmetic operations over a (multi)set of positive numbers (addition, multiplication, or a combination of them). The main advantage of using variable P systems consists in the small number of used rules regardless how many numbers are involved in the operation: e.g., the addition requires only 3 rules, while the multiplication only 27 rules.

Approach Based on the Ordered Fuzzy Decision Making System Dedicated to Supplier Evaluation in Supply Chain Management.

The selection of suppliers represents a pivotal aspect of supply chain management and has a considerable impact on the success and competitiveness of the organization in question. The selection of a suitable supplier is a multi-criteria decision making (MCDM) problem based on a number of qualitative, quantitative, and even conflicting criteria. The aim of this paper is to propose a novel MCDM approach dedicated to the supplier evaluation problem using an ordered fuzzy decision making system. This study uses a fuzzy inference system based on IF-THEN rules with ordered fuzzy numbers (OFNs). The approach employs the concept of OFNs to account for potential uncertainty and subjectivity in the decision making process, and it also takes into account the trends of changes in assessment values and entropy in the final supplier evaluation. This paper's principal contribution is the development of a knowledge base and the demonstration of its application in an ordered fuzzy expert system for multi-criteria supplier evaluation in a dynamic and uncertain environment. The proposed system takes into account the dynamic changes in the value of assessment parameters in the overall supplier assessment, allowing for the differentiation of suppliers based on current and historical data. The utilization of OFNs in a fuzzy model then allows for a reduction in the complexity of the knowledge base in comparison to a classical fuzzy system and makes it more accessible to users, as it requires only basic arithmetic operations in the inference process. This paper presents a comprehensive framework for the assessment of suppliers against a range of criteria, including local hiring, completeness, and defect factors. Furthermore, the potential to integrate sustainability and ESG (environmental, social, and corporate governance) criteria in the assessment process adds value to the decision making framework by adapting to current trends in supply chain management.

PENDAMPINGAN BELAJAR SISWA SEKOLAH DASAR UNTUK MENINGKATKAN KEMAMPUAN CALISTUNG

This study focuses on the learning assistance at Rumah Baca Loloan aimed at improving the reading, writing, and arithmetic (calistung) skills of elementary school students. The background of this research is the concern over the low calistung abilities among elementary school students, which could potentially hinder their academic performance in the future. This assistance program is implemented in response to the urgent need to improve basic skills through a more personalized and supportive learning approach. The assistants, consisting of trained PGSD students, pay special attention to the students, ensuring they receive the necessary help to overcome learning difficulties. The results of the assistance show a significant improvement in the students' calistung abilities after participating in this program. Indicators of improvement include increased reading fluency and text comprehension, improved writing skills with correct spelling and sentence structure, as well as increased speed and accuracy in basic arithmetic. Additionally, there is an increase in students' motivation to learn and self-confidence. The significance of these findings is that learning assistance outside of formal school hours can play an important role in enhancing students' basic skills. Support from parents and the community also proves to be crucial in the success of this program. The importance of this service lies in the evidence that structured and sustained interventions can have a significant positive impact on students' academic development, highlighting the need to expand similar programs to reach more students in need.

A randomized controlled study of a second grade numeracy intervention with Swedish students at-risk of mathematics difficulties.

Early numeracy interventions including basic arithmetic are crucial for young students at risk for early mathematics difficulties (MDs), yet few studies have evaluated numeracy interventions in second grade with a randomized controlled design. This pre- and post-test randomized controlled study evaluated the effects of an intensive 9-week numeracy and arithmetic programme for second-grade students at risk for early MDs. The focus of the programme was students' foundational understanding of numbers and mathematical concepts and procedural fluency with arithmetic tasks. A total of 753 first-grade students from 21 schools in Sweden were screened for low achievement in number knowledge and arithmetic. Students considered at risk for MDs (≤25 percentile on two consecutive first-grade mathematics screenings) were individually randomized to an intervention group (n = 32) or control group (n = 30) at the beginning of second grade (7-8 years old). Trained teachers administered the one-to-one, explicit programme to intervention group students in elementary school settings. The intervention group received numeracy instruction emphasizing foundational mathematics concepts and procedures. Controls received teaching as usual with potential special education support provided by their schools. The intervention group demonstrated significantly greater improvements in conceptual knowledge, arithmetic calculations and problem-solving compared to the control group, with medium size effects observed. A supplemental and intensive programme, with explicit instruction emphasizing numeracy, substantially improved knowledge and skills essential for arithmetic learning. Instruction in conceptual number knowledge and procedures also shows a significant impact on basic arithmetic problem-solving.

Weaker groupitizing abilities in magnitude perception in dyscalculia

The mechanisms underlying simple arithmetic difficulties in dyscalculia are unknown. Evidence suggests deficiencies may arise from basic processes supporting magnitude perception and arithmetic proficiency. We focused on ‘groupitizing’, the ability to decompose and recompose quantities into a whole, examining whether adults with dyscalculia could benefit from visuospatial grouping cues when asked to estimate the quantity of a given set of different number ranges. Results showed adults with dyscalculia tended to focus on individual items rather than grouping them, resulting in low estimation abilities in both small and large quantity ranges. However, the results highlight the importance of differentiating between large and small quantities in the groupitizing process: only where arithmetic was easy (small but not large quantities), were adults with dyscalculia able to demonstrate significant improvement when given grouping cues. Our findings suggest weaker groupitizing abilities play a significant role in magnitude perception and arithmetic proficiency levels in dyscalculia. Educational relevance statementPeople with dyscalculia have difficulties in basic arithmetic that impact their life in many ways, from simple tasks such as calculating change, to life-changing events such as entering higher education or making financial decisions. They use immature counting strategies to solve simple arithmetic questions despite otherwise intact intellectual abilities. The reasons for this are unclear. We argued there may be a link between simple arithmetic proficiency and quantity perception processes. Our study investigated the mechanisms underlying quantity perception that play a role in dyscalculia deficiencies. We focused on ‘groupitizing,’ a process that enables a person to estimate a given quantity more efficiently by recomposing subgroups rather than simply using a counting strategy. The findings suggested a disadvantage in basic groupitizing in people with dyscalculia. Moreover, counting procedures interfered with groupitizing processes and hindered arithmetic proficiency.

Analogy and Generalization as a Driving Force of Learning Mathematics – the Case of a Matrix Analog of a Zero of a Polynomial

The concepts of polynomials and matrices essentially expand and enhance the elementary arithmetic of numbers. Once introduced, polynomials and matrices open up new interesting mathematical challenges which extend to new fields of mathematical explorations within university mathematics. We present an aspect of a rather elementary exploration of polynomials and matrices, which offers a new perspective and an interesting matrix analog to the concept of a zero of a polynomial. The discussion offers an opportunity for better comprehension of the fundamental concepts of polynomials and matrices. As an application, we are led to the meaningful questions of linear algebra and to an easy proof of the otherwise advanced and abstract Cayley–Hamilton theorem.

Extension of Vedic mathematics sutras for square calculation

Abstract Square is one of the basic arithmetic operations. Referring to the earlier contributions provided by ancient Indian mathematicians concerning square calculation, various techniques have been suggested by Arybhata, Bhaskara I, Sridhara, Mahavira and Bhaskara II to quote a few. Besides, P. Swami Bharati Krishna Tirthji discussed various techniques for square calculating based on various sutras. In this paper, we summarise the works done by ancient Indian mathematicians and discuss the applications of calculating squares using different sutras. In addition, we propose a new method of calculating square based on the drawbacks identified and that can be applied for the square calculation of any number. Our method splits the number and hence makes the calculation simpler. Moreover, for bigger values, nested calculations can also be applied.

Unlocking mathematical marvels: Exploring number patterns in the Rangku Alu traditional game

Ethnomathematics, an interdisciplinary field at the intersection of cultural studies and mathematics, examines the intricate mathematical practices found within diverse cultural settings. This research focuses on Rangku Alu, a traditional game from Indonesia, to explore its embedded mathematical elements. Using a qualitative ethnographic methodology, the study uncovers the mathematical concepts within the game's cultural context. It traces the origins and cultural importance of Rangku Alu in Manggarai, East Nusa Tenggara, through combining participant observation, interviews, and literature review. The game, which prominently features bamboo, is a staple activity in harvest celebrations and serves as a way to express gratitude. During the game, players engage in synchronized movements that create rhythmic patterns and interactive sequences. The analysis reveals seven distinct numerical patterns, derived from the strikes of bamboo, jumping sequences, and player formations. These patterns include both odd and even sequences, as well as basic arithmetic operations. Rangku Alu not only entertains but also provides a unique setting for exploring numerical concepts. This study highlights the potential of integrating traditional practices like Rangku Alu into mathematics education to promote inclusivity and enhance mathematical understanding in multicultural environments.

Demetriou’s tests and levels of algebraic abilities and proportional reasoning in seventh, eighth, and ninth grades

Developing algebraic thinking is a key factor in learning mathematics. Despite its importance, many students still struggle with algebraic concepts. This research investigates students’ achievements in algebraic thinking using Demetriou’s test across 7<sup>th</sup> (approximately 12-13 years old), 8<sup>th</sup> (approximately 13-14 years old), and 9<sup>th</sup> (approximately 14-15 years old) grades. The study analyzes performance in different levels of algebraic tasks (i.e., [1] extrapolation of relationships, [2] coordinating simple structures, [3] operating with undefined symbolic structures, and [4] coordination with undefined structures), revealing a significant developmental leap in algebraic abilities during the 9<sup>th</sup> grade. While no statistically significant differences were found in the first level, 9<sup>th</sup> grade students demonstrated superior performance in levels 2, 3, and 4, suggesting cognitive readiness for abstract algebraic concepts around the age of 14. The research unveils a disjointed development in algebraic abilities, indicating a progression from basic arithmetic operations to proportional reasoning before the full integration of algebraic thinking. Notably, tasks involving variables in the third level pose persistent challenges for students. The findings contribute to understanding the optimal age for introducing algebraic concepts and underscore the importance of considering cognitive development in mathematics education. The study proposes implications for educators, such as emphasizing proportional reasoning in earlier grades and employing differentiated instruction based on individual students’ abilities.

Collatz Conjecture Solution

The Collatz conjecture, also known as the 3n + 1 conjecture, stands as one of the most enduring unsolved problems in mathematics. Proposed by German mathematician Lothar Collatz in 1937, the conjecture poses a deceptively simple question: can iteratively applying two basic arithmetic operations—dividing even numbers by 2 and multiplying odd numbers by 3 and adding 1—lead any positive integer to converge to the value 1? Despite extensive computational verification for vast ranges of integers, a proof or disproof remains elusive. This abstract explores the historical background, various attempts at proving the conjecture, and its implications across mathematical domains.