Algebraic Reasoning Research Articles (Page 1)

In order to improve students’ conceptual understanding and critical thinking skills in mathematics education, this study looks at the problem-solving approach as an instructional tool. Mathematics is a content-focused subject that requires students to think, evaluate, and even apply concepts in the real world. It cannot be reduced to simply memorizing formulas and following algorithms. However, the predominant teaching approach in Pakistan, like in many other school systems, is rote memorization, which leaves little room for inquiry or higher order thinking. The aim of this study was to investigate whether employing problem-solving techniques could serve as a “bridge” between these levels, allowing senior secondary school pupils to learn more deeply. Three hundred Grade 9 students from public and private schools in three districts in southern Punjab participated in a quasi-experimental study using a pretest-posttest control group design. After being split up into two groups, the subjects studied the same material, with the first group using a problem-solving approach and the second group listening to a traditional lecture. Two standardized tools were used to collect the data: the Mathematical Critical Thinking Assessment (MCTA), which consists of 20 context problems to measure skills in appropriate reflection of thinking, evaluation, and decision making, and the Conceptual Understanding Test (CUT), which consists of 25 test items to assess algebraic reasoning and understanding. Cronbach’s α values for both instruments were 0.87 and 0.89, respectively, indicating good reliability. The study was conducted for 8 weeks, the experiment group received RMT, collaborative discussion, reflexive task while the control group followed conventional textbook teaching method. Statistical analysis The data were analyzed using the SPSS Version 26 software which employed descriptive statistics, independent and paired sample t-test, (ANOVA) and Pearson correlation test. The experimental and control groups had significantly different (t = 9.52, p <. 001) and a mean of 18.6 point improvement. Additionally, we found a robust positive association (r = 0.71, p <. 001) between CT and conceptual understanding was found, indicating that they are interrelated. These results show that problem-solving has a positive impact on conceptual understanding, logical reasoning and critical reflection abilities of students. The method not only increases academic achievement but also teaches students how to think critically and creatively. This study is in favor of constructivist theory, which claims that learners do not receive passively knowledge, but rather they achieve it interactively and reflectively. Apart from this, it also has the following practical implications for mathematics educators, administrators and curriculum developers : problem-solving pedagogies should be built in to curricula of mathematics educations as well as teacher education. The author argues that problem-solving approach is ideal teaching and learning strategy because it triggers curiosity, makes learning experiences meaningful, and also help students acquire metaphysical reasoning and mental tools needed in the twenty-first century.

This systematic literature review (SLR) investigates transformative teaching strategies for enhancing students’ algebraic thinking, a foundational competency in mathematics education. Recognizing the long-term significance of algebraic reasoning, this review synthesizes findings from 25 peer-reviewed articles published between 2022 and 2024, identified through comprehensive searches in Scopus and Web of Science, and selected using the preferred reporting items for systematic reviews and meta-analyses framework. The sample size aligns with accepted standards for SLRs in emerging educational domains. Drawing from recent scholarship, this study constructs a structured framework across four interrelated themes: (1) cognitive transitions in algebra learning, illuminating how students shift from arithmetic to abstract reasoning; (2) innovative pedagogical strategies that promote active, student-centered learning; (3) representational fluency, highlighting the role of visual, symbolic, and contextual tools in bridging conceptual gaps; and (4) developmental alignment in curriculum and assessment design, advocating for instructional sequencing tailored to learners’ cognitive growth. The synthesis reveals that integrating cognitive, pedagogical, and curricular dimensions significantly strengthens algebraic reasoning. Despite its methodological rigor, the study is limited to English-language journal articles and excludes grey literature, which may constrain the comprehensiveness of findings. Moreover, the literature reflects developments only up to 2024, and more recent innovations may not be captured. Nonetheless, this review contributes a timely, evidence-based model for guiding instructional reform in algebra education and underscores the need for targeted, flexible strategies that support students’ conceptual progression toward higher-level mathematics.

Mathematical reasoning, the cognitive process of drawing logical conclusions and solving problems based on mathematical concepts, is fundamental for academic success and lifelong learning. This study investigated the proficiency levels in mathematical reasoning among Grade 10 students in a private secondary school in Bacolod City, Philippines. Focusing on four domains: algebraic, geometric, proportional, and statistical reasoning, the study employed a quasi-experimental, one-group pretest–posttest design. A total of 215 students initially participated in the pre-test, after which the section with the lowest average score underwent a two-week intervention using Learning Activity Sheets (LAS). The same assessment tool was administered as a post-test to measure improvements. Quantitative results revealed significant gains in algebraic, geometric, and proportional reasoning, with geometric reasoning showing the most notable improvement. Statistical reasoning improved modestly, remaining the most challenging domain for many students. Student feedback further supported the effectiveness of the LAS, highlighting improved comprehension, structured problem-solving, and better recall of key concepts. The study concluded that well-designed LAS interventions can effectively improve students’ reasoning skills and may serve as a model for classroom-based instructional strategies.

<p class="Afiliasi">Algebra is a fundamental topic in the mathematics curriculum at Madrasah Tsanawiyah (MTs), yet many students continue to face challenges in solving algebraic problems. This study aims to examine the algebraic thinking skills of eighth-grade students at MTs Unggulan PP Amanatul Ummah Surabaya, particularly through number-oriented and structure-oriented approaches. A descriptive qualitative method was employed, involving 61 students as participants. The instruments included an algebraic thinking test incorporating visual illustrations and in-depth interviews with six selected students representing high, medium, and low ability levels. The findings reveal that only high-ability students were able to effectively apply structure-oriented algebraic thinking. The majority of students demonstrated moderate-level skills, with primary difficulties in constructing mathematical models based on problem structures. These results underscore the importance of instructional strategies that cultivate students’ structural reasoning in algebra. The study provides valuable insights for educators to design learning activities that support the development of students' algebraic thinking, and it serves as a reference for future research to explore targeted interventions aimed at enhancing algebraic problem-solving abilities in MTs students.</p>

This study investigated the misconceptions held by first-year university students regarding basic mathematical concepts, with a focus on integer operations, fractions, and elementary algebra. Conducted at a private university in Aceh, Indonesia, the research involved 26 students enrolled in the Elementary Algebra course. A diagnostic test consisting of five open-ended questions was administered, followed by structured interviews with 6 students who exhibited notable errors. The findings revealed persistent misconceptions, including misapplication of integer rules, incorrect fraction operations (e.g., adding numerators and denominators directly), and flawed algebraic reasoning. These issues indicate a reliance on procedural knowledge rather than conceptual understanding, often rooted in prior learning experiences. The results highlight the importance of early diagnostic assessments and concept-focused instruction in bridging the gap between school and university mathematics. Addressing these misconceptions is essential to support students’ mathematical development and academic success in higher education.

This action research investigates the effectiveness of LEGO® manipulatives in improving the algebraic reasoning skills of at-risk grade 8 learners. The study aims to determine if incorporating hands-on activities with Lego bricks leads to statistically significant gains in students’ understanding and applying algebraic concepts. A purposive sample of 8 grade 8 students identified as at-risk in mathematics participated in the study. Selection criteria included past academic performance, teacher observations of learning difficulties, and documented learning challenges. This research employed an action research design with iterative cycles of implementation, observation, reflection, and refinement. The intervention involves integrating LEGO® manipulatives into lessons targeting key algebraic concepts. These activities will promote a deeper understanding of mathematical relationships and problem-solving skills. Pretest and post-test were administered, assessing algebraic reasoning skills to measure quantitative changes. Additionally, qualitative data will be gathered through classroom observations, student interviews, and teacher reflections. This data triangulation will provide a holistic understanding of the interventions’ impact. According to the results, the post-test scores of students who participated in the Lego manipulatives intervention improved statistically significantly compared to their pretest levels. Qualitative data from observations, interviews, and reflections reveal increased student engagement, improved conceptual understanding, and enhanced problem-solving skills when utilizing LEGO® manipulatives. This action research can potentially demonstrate the effectiveness of LEGO® manipulatives in fostering algebraic reasoning skills among at-risk students. Developing more engaging and successful methods for teaching algebra to at-risk students may result in positive outcomes, which will also provide essential insights into mathematics education and inspire instructional approaches that accommodate a variety of learning styles. However, limitations inherent to action research, such as the small sample size and potential for research bias, necessitate further investigation in larger-scale studies.

Algebra is considered one of the most important parts of Mathematics teaching and learning, because it lays the foundations of abstract thinking as well as reasoning abilities among the lower secondary school pupils who have just transited from the world of numbers and computations to the area of equalities, signs, symbols and letters. The present article focuses on the fact that how the transition from arithmetic to algebra can be made more smooth. We have concentrated our experiments towards the approach of algebraic reasoning and its utilities in filling the gap between arithmetic and beginning algebra in lower secondary school education.We also underline the importance of another approach in overcoming the challenges in the transition from arithmetic to algebra, to enhance and make algebraic learning more effective, with special considerations to word problem-solving processes. In our opinion, we have to go through three phases in the introducing of algebra in Grade 7 Mathematics education: Regula Falsi method (based only on numerical calculations); functional approach to algebra (which combines the numerical computation with letter-symbolic manipulation); and writing equations to word problems. The conclusions of the present article would be helpful to Mathematics teachers for applying themselves to develop the pupils’ interest in word problem-solving processes during algebra teaching classroom activities. Subject Classification: 97B10, 97C30, 97C50, 97D10, 97D40

Algebra is a fundamental component of mathematics education that supports students’ success in higher-level mathematical learning. However, many students struggle to understand algebraic concepts due to their abstract nature and the limited use of effective instructional strategies. This scoping review aims to (1) identify how algebra teaching activities are represented in the literature, (2) explore the main instructional themes and approaches discussed, and (3) examine existing research gaps that require further study. A total of 143 studies published between 2015 and 2025 were retrieved from the Scopus database, and after a rigorous screening process, six studies met the inclusion criteria for full analysis. The review identified five dominant instructional themes: contextual problem-based learning, visual and concrete representations, technology-enhanced and game-based learning, collaborative discussions, and reflective reasoning. These approaches were found to enhance students’ engagement, conceptual understanding, and algebraic reasoning. Nevertheless, significant gaps remain in early algebra instruction, teacher professional development, and the integration of emerging technologies in classroom practice. The findings underscore the need for innovative, inclusive, and technology-integrated teaching strategies to strengthen students’ algebraic thinking and improve the quality of mathematics education.

Abstract Algebra is a core aspect of mathematics, often functioning as a gatekeeper to further studies in mathematics. Although a well-researched area, we still do not know how students’ algebraic reasoning can vary, including the understanding of the roles of various mathematical arguments. In an explorative study, using semi-structured, non-participant observations and Interpersonal Process Recall interviews, we analyse eight upper secondary students’ collective mathematical reasoning when solving algebraic tasks about arithmetic sequences. The results show that the majority of expressed arguments were anchored in relevant mathematical properties covering a wide spectrum of algebraic reasoning. The results indicate that it is in the first instance of the reasoning, the task situation, where the students interpreted the pattern differently, where the biggest variation of different aspects of algebraic reasoning was displayed. In addition, the identifying arguments constituted the main part of all expressed arguments, indicating that the core part of the reasoning was in the interpretation of the task. There were few arguments about the choice of strategy and its implementation, signalling that once an interpretation was made and agreed upon, the strategy choice did not have as dominant role as previous research has suggested. In most cases, the arguments provided for the conclusion, evaluative arguments, were implicit and connected with previously expressed identifying arguments. The results also show that identifying arguments was connected to the mathematical content of the task, whereas the difference in algebraic reasoning appears depends on students' solution constructions and their degree of conventional syntax.

This study develops a novel framework for bipolar-valued intuitionistic fuzzy positive implicative ideals (BPVIFPIIs) in BCK-algebras by integrating bipolar-valued intuitionistic fuzzy set theory with algebraic structures. The primary objective is to define and explore the properties of BPVIFPIIs in BCK-algebras, providing rigorous theoretical foundations supported by illustra-tive examples. Key conditions under which a bipolar-valued intuitionistic fuzzy set qualifies as a BPVIFPII are established. The findings reveal significant connections between BPVIFPIIs and other fuzzy ideals, highlighting their role in advancing the understanding of uncertainty and algebraic reasoning. This research opens avenues for further exploration of bipolar fuzzy structures in algebra and their practical implications in decision-making processes involving uncertain data.

Cognitive processes are procedures for using existing knowledge to combine it with new knowledge and make decisions based on that knowledge. This study aims to identify the cognitive structure of students during information processing based on the level of algebraic reasoning ability. This type of research is qualitative with exploratory methods. The data collection technique used began by providing a valid and reliable test instrument for algebraic reasoning abilities for six mathematics education student programs at the Islamic University of Sultan Agung Indonesia. Subjects were selected based on the level of upper, middle, and lower algebraic reasoning abilities. The results showed that (1) students with the highest level of algebraic reasoning ability meet the logical structure of Logical Reasoning which shows that students at the upper level can find patterns and can generalize; (2) Students at the intermediate level understand the cognitive structure of Symbolic Representations, where students can make connections between knowledge and experience and look for patterns and relationships but have difficulty making rules and generalizations; (3) students at lower levels understand the cognitive structure of Comparative Thinking, where students are only able to make connections between prior knowledge and experience.

Few studies have examined the affective factors influencing students' algebraic reasoning. This study aims to investigate the effect of student engagement and self-regulated learning on algebraic reasoning. The research employed a quantitative correlational design. Using a cluster sampling technique, 202 students from Islamic State junior high schools in Mataram were selected as participants. Data were collected through tests and questionnaires. The instruments used included an algebraic reasoning test, student engagement questionnaires, and self-regulated learning questionnaires. Data analysis involved both descriptive analysis (categorical and statistical) and inferential analysis (prerequisite tests and hypothesis testing). The results of this study indicate that student engagement has a significant influence on algebraic reasoning (t = 2.418, p = 0.017 < 0.05). However, self-regulated learning did not show a significant effect on algebraic reasoning (t = -0.425, p = 0.671 > 0.05). Additionally, student engagement and self-regulated learning, when considered together, significantly influence algebraic reasoning (F = 3.117, p = 0.046 < 0.05). The study also found that student engagement and self-regulated learning account for 3% of the variance in algebraic reasoning (R² = 0.03), with the regression equation Y = 43.491 + 0.277X₁ - 0.06X₂. These findings suggest that teachers should prioritize fostering student engagement and self-regulated learning in the classroom, emphasizing interactive, collaborative, and contextually relevant algebra instruction. Abstrak: Beberapa penelitian telah mengeksplorasi faktor afektif yang memengaruhi penalaran aljabar siswa. Penelitian ini bertujuan untuk mengetahui pengaruh keterlibatan siswa dan pembelajaran yang diatur sendiri terhadap penalaran aljabar. Jenis penelitian yang digunakan adalah kuantitatif korelasional. Dengan teknik sampling klaster, 202 siswa dari Sekolah Menengah Pertama Negeri di Mataram dipilih sebagai sampel penelitian. Data dikumpulkan menggunakan tes dan kuesioner. Instrumen yang digunakan dalam penelitian ini adalah tes penalaran aljabar, kuesioner keterlibatan siswa, dan kuesioner pembelajaran yang diatur sendiri. Analisis data yang digunakan dalam penelitian ini adalah analisis deskriptif (kategori dan statistik deskriptif) dan analisis inferensial (uji prasyarat dan uji hipotesis). Hasil penelitian ini menunjukkan bahwa keterlibatan siswa berpengaruh terhadap penalaran aljabar (t = 2,418, p = 0,017 < 0,05). Namun, pembelajaran yang diatur sendiri tidak berpengaruh terhadap penalaran aljabar (t = -0,425, p = 0,671 > 0,05). Selain itu, keterlibatan siswa dan pembelajaran yang diatur sendiri secara bersamaan berpengaruh terhadap penalaran aljabar (F = 3,117, p = 0,046 < 0,05). Penelitian ini juga menunjukkan bahwa keterlibatan siswa dan pembelajaran yang diatur sendiri memberikan kontribusi sebesar 3% (R² = 0,03) terhadap penalaran aljabar dengan persamaan regresi Y = 43,491 + 0,277X₁ - 0,06X₂. Temuan ini memberikan implikasi bagi guru untuk memprioritaskan keterlibatan siswa dan pembelajaran yang diatur sendiri di dalam kelas, dengan menekankan pembelajaran aljabar yang interaktif, kolaboratif, dan relevan dengan konteks.

This paper starts from the hypothesis that algebraic reasoning can be used as an axis between different mathematical domains at school. This is relevant given the importance attributed to mathematical connections for curriculum development and the algebraic reasoning makes it possible to articulate it in a coherent manner. A definition of generalized algebraic reasoning is proposed, based on the notion of elementary algebraic reasoning of the onto-semiotic approach, and it is used to highlight the presence of typical algebraic processes in problem solving in geometrical contexts. To develop these ideas, a training course is designed and implemented with in-service secondary school teachers. Based on design-based research, the results obtained are contrasted with the expected answers. In this way, relevant information is obtained on how teachers mobilize different typically algebraic processes, that is, particularization-generalization, representation-signification, decomposition-reification and modelling. Actually, it is clear to affirm that teachers need specific training to improve their skills about how algebraic reasoning can help them to develop mathematical connections with their students.

Many important computational structures involve an intricate interplay between algebraic features (given by operations on the underlying set) and relational features (taking account of notions such as order or distance). This paper investigates algebras over relational structures axiomatized by an infinitary Horn theory, which subsume, for example, partial algebras, various incarnations of ordered algebras, quantitative algebras introduced by Mardare, Panangaden, and Plotkin, and their recent extension to generalized metric spaces and lifted algebraic signatures by Mio, Sarkis, and Vignudelli. To this end, we develop the notion of clustered equation, which is inspired by Mardare et al.'s basic conditional equations in the theory of quantitative algebras, at the level of generality of arbitrary relational structures, and we prove that it is equivalent to an abstract categorical form of equation earlier introduced by Milius and Urbat. Our main results are a family of Birkhoff-type variety theorems (classifying the expressive power of clustered equations) and an exactness theorem (classifying abstract equations by a congruence property).

Controversial issues have the potential to stimulate differences or cognitive conflicts in problem-solving, both in terms of approach, argumentation, and the solutions produced. Resolving cognitive conflict can be done not only through conventional logical thinking but also by understanding objects that encompass facts, concepts, principles, and skills, as well as considering external aspects surrounding the problem. This study aims to explore the controversy reasoning problems of university students in solving multiple representation problems. This research is a case study with a qualitative approach. The selected subjects were students who experienced controversy in solving multiple representation problems. The instruments used were tests and interviews. Data analysis was carried out through data reduction, data presentation, and conclusion drawing. There are three levels of controversial reasoning: clarification level, exploration level, and initial level. Among 99 students solving multiple representation problems, the results showed that 19 students (19.19%) were at the initial level, 45 students (45.45%) at the exploration level, and 35 students (35.35%) at the clarification level. Qualitatively, controversial reasoning at the clarification level is the best compared to the exploration and initial levels. The advantage of subjects at the clarification level is that, besides understanding the problem, their algorithmic skills are highly structured. These subjects are also able to provide concise, logical arguments and find the correct solution.

Novel methodology for targeting the optimal reactor and operating parameters based on the chemical system’s overall performance within catalyst lifecycle

Justification serves as a tool used to improve students' ability to understand mathematics and their proficiency in working on mathematical problems. However, despite its significance, student justification in the problem-solving process has not become a priority for teachers based on several studies. While justification research related to problem-solving has begun to develop, it is only limited to validating the truth of a mathematical solution. Thus, this qualitative descriptive research aims to analyze students' justification process in solving reasoning problems regarding the types of justification (interpretation, elaboration, prediction, and validation) and the function of each type. The research subjects consisted of two high school students in Indonesia with high abilities who solved algebraic mathematics reasoning problems. Meanwhile, data collection and analysis used the results of students solving algebraic reasoning problems involving the nature of justification tasks. The results show that the types of justification indicate a crucial role in students' problem-solving process. Apart from that, each of these also has the potential for use in the problem-solving process. Furthermore, this article also suggests several points that can be applied to develop student justification in reasoning algebraic problems.

Abstrak Penalaran aljabar menjadi topik hangat dalam dua dekade terakhir, tetapi siswa sekolah menengah mengalami kesulitan dalam mempelajari hal ini, sebab mengalami transformasi dari pemikiran aritmatika ke pemikiran aljabar yang abstrak. Penelitian ini menganalisis artikel penalaran aljabar dari basis data Scopus berdasarkan tahun penelitian, metode, dan sebaran geografis dalam satu dekade. Penelitian disajikan melalui tinjauan literatur sistematis dengan menggunakan protokol PRISMA meliputi Preferred Reporting Item for Systematic Reviews and Meta-Analytic. Dari proses seleksi, didapatkan 29 artikel yang memenuhi kriteria inklusi dan lolos tiga tahap penyaringan. Penelitian dengan topik ini cenderung mengalami penurunan, terutama dalam 4 tahun terakhir. Metode yang paling sering digunakan adalah metode penelitian kualitatif karena dianggap peneliti sangat cocok untuk meneliti kemampuan berpikir. Dari 8 wilayah yang melakukan penelitian penalaran aljabar, Indonesia merupakan negara yang paling banyak melakukannya. Hal ini dilatarbelakangi rendahnya penalaran aljabar siswa. Peluang penelitian pada topik ini masih terbuka lebar dengan menggunakan penelitian ini sebagai rujukan. Abstract Algebraic reasoning has become a hot topic in the last two decades, but high school students have difficulty in learning it, because they experience a transformation from arithmetic thinking to abstract algebraic thinking. This study analyzes algebraic reasoning articles from the Scopus database based on the year of research, methods, and geographical distribution in one decade. The research is presented through a systematic literature review using the PRISMA protocol including the Preferred Reporting Item for Systematic Reviews and Meta-Analytic. From the selection process, 29 articles were obtained that met the inclusion criteria and passed three stages of screening. Research on this topic tends to decline, especially in the last 4 years. The most frequently used method is the qualitative research method because researchers consider it very suitable for researching thinking skills. Of the 8 regions that conduct algebraic reasoning research, Indonesia is the country that does it the most. This is due to the low algebraic reasoning of students. Research opportunities on this topic are still wide open by using this research as a reference.

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.

<p style="text-align:justify">Scaffolding dialogue is a concept in learning that refers to the support or assistance given to individuals during the dialogue process. The main objective of this research is to create a basic structure of dialogue to help and support students during the learning process in improving their algebraic reasoning skills. Algebraic reasoning is a process in which students generalize mathematical ideas from a certain set of examples, establish these generalizations through argumentative discourse, and express them in a formal and age-appropriate way. The study was designed using the grounded theory qualitative model method, which used three sequential steps: open coding, selective coding, and theoretical coding. The research was conducted on students of the mathematics education department at Universitas Islam Sultan Agung. Data collection methods include algebraic reasoning ability tests, questionnaires, and interviews. Data analysis in grounded theory is an iterative and non-linear process that requires researchers to constantly move back and forth between data collection and analysis. This process aims to produce a theory that is valid and can explain phenomena well based on empirical data obtained during research. The dialogue scaffolding strategy framework in improving students' algebraic reasoning abilities includes instructing, locating, identifying, modeling, advocating, exploring, reformulating, challenging, and evaluating.</p>