Algebraic Reasoning Research Articles

ANALISIS HAMBATAN BELAJAR (LEARNING OBSTACLE) SISWA SMP TERHADAP PENALARAN ALJABAR

This study aims to identify learning barriers experienced by junior high school students in understanding algebraic concepts and analyze how these barriers affect students' ability to solve number pattern problems. Indicators of epistemological barriers in the study are barriers to understanding concepts, barriers in applying procedures and barriers in operations. This type of research uses a qualitative approach with a case study type of research. This research was conducted at SMPN 3 Labuapi in the 2023/2024 school year. The instruments used in this study were descriptive test questions, interviews and documentation on students of class VIII-A SMPN 3 Labuapi. The subjects in this study were 6 students of class VIII-A SMPN 3 Labuapi. The instrument used in this research is a description test question as many as two questions and conducting interviews with each subject. Interviews were conducted to find out student obstacles in algebraic reasoning on number pattern material. The results of the analysis show that there is an epistemological obstacle in the 1 students of SMPN 3 Labuapi, namely errors in determining formulas, theorems or definitions and also errors in calculating values and the steps for solving number pattern problems that are ordered are not appropriate and errors in writing formulas that cause errors in calcu[i]lations. And the results of this study show that there are still many students experience epistemological barriers in the conceptual barriers section, procedural barriers and operational technique barriers.

Dialogic Scaffolding: How to Design Critical Questions in Developing Students Algebraic Reasoning?

Dialogic Scaffolding: How to Design Critical Questions in Developing Students Algebraic Reasoning?

Algebraic reasoning of high school students in solving inverse function problems: Viewed from mathematical resilience

Algebra is one of the essential branches of mathematics that often presents challenges for students, especially when faced with inverse function problems. This study aims to describe algebraic reasoning in solving inverse function problems from the perspective of students' mathematical resilience. This qualitative research uses a case study design involving 33 eleventh-grade students from a public high school in West Bandung Regency. The instruments used were a mathematical resilience scale and an algebraic reasoning test. Data analysis was conducted through data reduction, presentation, and conclusion drawing. The results showed that 15.15% of students had low resilience, 72.72% had moderate resilience, and 12.12% had high resilience. There were different characteristics in algebraic reasoning among students in these resilience categories. The implications indicate that targeted learning strategies can enhance algebraic reasoning, especially for moderate and low-resilience students.

Exploring Pointed Sets and their Role in the Category of Pointed F-Sets: A Study on Kernels in the Category F-Sets

This paper delves into the realm of pointed sets and their significance within the framework of algebraic structures, particularly focusing on their role in the category of pointed F-sets. Pointed sets, characterized by a single nullary operation identifying a base point, serve as fundamental components in algebraic reasoning, with applications in various mathematical domains. The category of pointed F-sets, comprising pointed sets as objects and pointed F-associations as morphisms, offers a rich ground for exploring the behavior of specialized functions. Central to our investigation is the demonstration of kernels within the category F-sets, elucidating the foundational properties and structural insights underlying pointed sets and their associations.

Expanded model for elementary algebraic reasoning levels

Expanded model for elementary algebraic reasoning levels

Kotobee-Assisted Means-End Analysis Learning: Impact on Algebraic Reasoning Abilities Reviewed From Students' Cognitive Stage

The Indonesian curriculum emphasizes tailored learning based on students' developmental stages and achievement levels, focusing on cognitive development, particularly in algebraic reasoning, which is very important for students' overall cognitive maturation. This study aimed to determine the effect of the implementation of the learning model and students' cognitive stage on algebraic reasoning ability. The study was conducted using quantitative research with a post-test design to achieve the research objective. Two classes were taken as samples from one of the high schools in West Bandung Regency. The instruments used were the Test of Logical Thinking (TOLT) to categorize students into students' cognitive stage groups and algebraic reasoning test instruments to determine the achievement of algebraic reasoning ability. Data analysis was done using statistical tests, including a normality test, homogeneity test, and two-way ANOVA test. The results are: there is an effect of implementing the Means-End Analysis (MEA) learning model supported by Kotobee on algebraic reasoning; there is an effect of students' cognitive stage on algebraic reasoning; and there is no interaction effect of students' cognitive stage and implementation learning model on algebraic reasoning ability. Further research on other mathematical skills using the Kotobee-based MEA learning model is needed.

Analysis of Students' Algebraic Reasoning Level in Learning Limit Function

Analysis of Students' Algebraic Reasoning Level in Learning Limit Function

Connecting the threads: the role of multiplicative thinking in algebraic, geometrical, and statistical reasoning

ABSTRACT Making connections within and between different aspects of mathematics is recognised as fundamental to learning mathematics with understanding. However, exactly what these connections are and how they serve the goal of learning mathematics is rarely made explicit in curriculum documents with the result that mathematics tends to be presented as a set of discrete, disconnected topics. Interest in establishing a more coherent approach to the teaching and learning of school mathematics has led to a focus on big ideas. That is, networks of related concepts, skills and ways of thinking that facilitate learning mathematics with understanding. Research on learning progressions has helped identify what these big ideas are and how they serve to build connections within and between different aspects of mathematics. This paper draws on research that provides an evidenced-based learning progression for multiplicative reasoning to illustrate the connective role of multiplicative thinking in the development of algebraic, geometrical, and statistical reasoning.

Algebraic Representations of a Splitting Task: Implications of Numerical Reasoning

ABSTRACT This qualitative research study uses middle-grades students’ numerical reasoning to model their symbolic representations of the relationship between two multiplicatively related unknowns on an algebra task. Students in sixth grade through ninth grade participated in clinical interviews that assessed their numerical reasoning using the Number Sequences framework, their interpretation of the equal sign, and their algebraic reasoning in the context of writing equations. Analysis used the mental structures that define their numerical reasoning to explain their algebraic capabilities, and changes in their reasoning about the equal sign in different contexts. Findings show that splitting, one mental structure related to the students’ numerical reasoning, is related to students’ symbolic representations of multiplicatively related unknowns. Furthermore, students who had not constructed a splitting operation tended to use numerical examples in place of reasoning about unknowns, and to reason operationally about the equal sign to compensate for this limitation on an algebra task.

Promoting Linear Algebraic Reasoning among Students: Affordances and Challenges

As students transition from the mathematics they learn in school years, including their first-year calculus courses, to the first course in linear algebra, they experience discontinuities in their perspective of what mathematics is. Their propensity to continue applying the same habits of learning in the face of this change leads to failure and frustration. The failure manifests itself in the quality of understanding basic concepts as well as in the lack of linear algebraic reasoning. Instructional treatments applied in my teaching experiments to foster students’ ability to reason linear algebraically resulted in mixed success – some of the treatments were successful, others less so. The latter are accounted for by the structural complexity of the subject matter and students’ background knowledge. The pedagogical approaches offered in this paper are oriented within a particular theoretical framework for the learning and teaching of mathematics, called DNR. Reflections and broader implications are addressed through the lenses of this framework.

Mathematical Symbols in Academic Writing: The Case of Incorporating Mathematical Ideals in Academic Writing for Education Researchers

Mathematical symbols, such as those embodying quantum concepts, are indispensable for conveying complex ideas and relationships in academic writing. However, some education researchers and students keep a distance from anything mathematical: algebraic equations, geometrical reasoning, or statistical symbols. How to lower the access threshold for this type of mathematical narrative and reveal the meanings of a range of quantum conceptions to modern educators thus becomes a real problem. Using the pendulum motion equation as a reference point, I argue in this article for the advantages of academic English or French writing genres that fuse a range of mathematical symbols of quantum concepts and conceptual change. Such writings help demonstrate how incorporating the idea of probability (a) refines the debate among conceptual, verbal, and mathematical academic writing; (b) allows new conceptions that draw on the insights from quantum cognition-supported theories; (c) helps explain students’ understanding of mathematical symbols; and (d) offers a new taxonomy for categorizing academic writings.

Students’ Algebraic Proficiency and Attitude towards Learning Algebra

Learning algebra is a difficult process that necessitates a positive attitude in order to maintain interest and master key concepts. Previous research found that students' attitudes influenced their algebra learning. As a result, more input is required to observe the current influence of attitudes on algebra learning. This study aims to investigate secondary school students’ levels of proficiency and attitudes towards learning algebra. A case study was employed to collect data using an algebra test, a questionnaire on attitude, and task-based interviews. The subjects of this study were Form Three and Form Four students who were in the Dual Language Programme (DLP) in a school in Malaysia. A total of 93 students volunteered to participate in this study, and three of them were interviewed for insights into solving algebra problems. This study revealed that (1) there was no significant difference in algebra achievement between the Form Three and Form Four students; (2) there was a negative correlation between the achievement and attitudes; and (3) the students displayed rather low proficiency in answering the test questions, which could be attributed to difficulties with comprehension of problems in algebra and inadequate reasoning when applying problem solving strategies. These findings have implications for developing positive attitudes towards the learning of algebra. It is recommended that educators look into these areas to equip students with the right attitudes and knowledge for advanced learning at university.

Effects of Geometer’s Sketchpad on Algebraic Reasoning Competency amongst Students in Malaysia

This study investigates the impact of incorporating Geometer's Sketchpad into GSP-based instruction on the algebraic reasoning proficiency of students. The research outlines the application of GSP-based instruction, utilizing Geometer's Sketchpad in Mathematics education, with a specific focus on solving quadratic problems. A quasi-experimental design, involving non-equivalent pre-test and post-test assessments, was conducted on 60 second-grade students at a private secondary school in Kuala Lumpur. The experimental group, consisting of 30 students, utilized Geometer's Sketchpad in their learning, while the control group, comprising 30 students, received traditional instruction. Data analysis was performed using ANATES 4 and SPSS 25.0 software, incorporating inferential statistics such as paired t-tests and one-way ANCOVA for quantitative data analysis. Both groups underwent an initial pre-test assessment. The research findings reveal a significant disparity in algebraic reasoning proficiency between the two groups, indicating substantial improvement after the intervention. Consequently, the integration of Geometer's Sketchpad in GSP-based instruction contributed to the enhancement of students' algebraic reasoning skills, particularly in quadratic problem-solving. In conclusion, this study underscores the potential of Geometer's Sketchpad as an instructional intervention, urging mathematics educators to contemplate its incorporation as an alternative teaching tool.

Algebraic thinking ability of grade VIII students using a Problem-Based Learning (PBL)

Students still have limited algebraic reasoning skills, so a learning model called Problem-Based Learning (PBL) is utilized as a teaching strategy to help student develop their algebraic thinking abilities. This study uses the learning model Problem-Based Learning (PBL) to the algebraic thinking skills of class VIII students. The method of research is descriptive qualitative research. The students in class VIII students of SMPN 1 Lahat participated in this study. Problem-Based Learning (PBL) worksheet in accordance with three algebraic thinking abilities: generational activities, transformational activities, and global, meta-level, and mathematical activities. Data for the study came from observations, interviews and written exams. Data from the results of observation sheets are used to see the emergence of indicators of students' algebraic thinking during learning with a total of 2 sessions; data from the results of written tests are used to categorize students' algebraic thinking abilities after learning, taking 2 students from each category, and interviews to clarify the results of students' answers. The findings of this study show that students possess a range of low, medium, and high levels of algebraic thinking ability. The result of the implementation learning model showed that 14 students were in the medium category and 17 students were in the low category.

Creation of Problems by Prospective Teachers to Develop Proportional and Algebraic Reasonings in a Probabilistic Context

To promote optimal learning in their students, mathematics teachers must be proficient in problem posing, making this skill a cornerstone in teacher training programs. This study presents a formative action in which pre-service teachers are required to create and analyze a problem involving proportional reasoning within a probabilistic context. For this problem, they must identify the objects and processes involved in its resolution, recognize the degree of algebraic reasoning implied and identify potential difficulties for students. Subsequently, they need to formulate and analyze a new problem with variation, which mobilizes higher-level algebraic activity. Results indicate that prospective teachers struggle to pose problems that engage proportional reasoning, as well as to identify in their analysis which elements of proportional and algebraic reasoning are present in their solutions. Despite this fact, a significant percentage of participants adequately modify the original problem to address higher levels of algebraic reasoning, identifying in these cases the new algebraic objects and potential difficulties that might arise as the degree of generalization required in the solution increases. The study concludes by underscoring the importance of training in problem posing to enhance the knowledge and competences of prospective teachers concerning proportional and algebraic reasoning.

Preservice teachers’ errors in the creation and extension of sequences using color tiles: An exploratory study

AbstractRecognizing, building, describing, extending, and analyzing patterns are key components of algebraic reasoning. Oftentimes pattern‐based instruction is used to support the understanding of functions and variables. In this study, elementary preservice teachers (n = 23) were asked to explore, extend, explain, create equations for, and analyze linear, geometric, and quadratic integer patterns using color tiles. Then they created, extended, described, and analyzed their own patterns while working in groups (n = 8). The research question was: What algebraic errors do PSTs demonstrate in the process of constructing, explaining, and analyzing their own linear, geometric, and quadratic patterns using color tiles? The analysis, which focused on errors and misunderstandings preservice teachers demonstrated in their final unit, revealed three specific areas of difficulties: figural errors, explanation challenges, and improper use of algebraic language. Participants failed to connect their algebraic images in a coherent manner, there was confusion in articulating their algebraic thought processes, and algebraic terminology was improperly used. Implications for teaching and further research are described.

Algebraic Reasoning About Timeliness

Algebraic Reasoning About Timeliness

Algebra Reasoning Ability Viewed From Student Gender Differences

Algebraic reasoning skills are important for students to master, with algebraic reasoning students are able to solve everyday problems in algebraic form. Gender is one of the factors that differentiates students' reasoning abilities in solving math problems. This study aims to describe algebraic reasoning in terms of gender differences. This type of research is qualitative with research subjects of four students taken from 31 students, Middle School class VIII by selecting two female students and two male students based on the average score level of each gender in solving System of linear equations of two variables questions based on indicator of algebraic reasoning ability. Data collection techniques in this study were tests, interviews and documentation. The results of this study indicate that the algebraic reasoning of female subjects is higher than that of male subjects. Some of the factors for the low algebraic reasoning of the subject include: the subject is not careful and forgets the formulas that have been taught.

Investigation of generalisation processes of secondary school students using multiple representations in a pattern task

The aim of this study is to show how the GeoGebra software-supported collaborative learning environment contributes to a group of students using multiple representations and reaching generalisations while solving problems collectively. Blending individual and collaborative learning activities were carried out with a learning method called ACODESA (from French: Apprentissage Collaboratif, Débat Scientifique, Auto-réflexion). Within this context, four 8th-grade students participating in the study were asked to solve the pattern problem by using the GeoGebra software and following the stages of the ACODESA method. The data were collected via a pattern problem developed by the researcher, students’ audio records and screen recordings, GeoGebra files and written productions. The data were analysed using Duval’s theory of registers of semiotic representation and Toulmin’s model. The Toulmin model was used to analyse students’ argument structures, that is, to determine their reasoning structures. The data analysis revealed that a GeoGebra software-supported collaborative learning environment supported secondary school students in using multiple representations, making algebraic reasoning and generalisation. The discussions in this process contributed to the students’ transformation of representations and the evolution of their representations. Providing students with the opportunity to express and examine the pattern with multiple representations, GeoGebra software helped students to obtain data, test their conjectures and thus reach generalisation.

Visual and symbolic representations as components of algebraic reasoning

Sixty (35 girls) ninth graders were assessed on measures of algebraic reasoning and usage of visual and symbolic representations (with a prompt for visual use) to solve equations and inequalities. The study grouped visual representations into two categories: arithmetic-visual, which entailed the use of real-world objects to represent specific values of variables, and algebraic-visual, which involved formal representations like the number line and the coordinate plane. Symbolic representations, on the other hand, encompassed the use of standard algorithms to solve equations, such as changing the place of terms in an equation. The results reveal that the use of algebraic visuals, as opposed to arithmetic visuals, was associated with enhanced algebraic reasoning. Further, although the students initially relied on standard algorithms to explain equations and inequalities, they could produce accurate algebraic-visual representations when prompted. These findings suggest that students have multiple representations of equations and inequalities but only express visual representations when asked to do so. In keeping with the general relationship between visuospatial abilities and mathematics, self-generated algebraic-visual representations partially mediated the relation between overall mathematics achievement and algebraic reasoning.