Algebraic Thinking Research Articles

PEMBANGUNAN UJIAN KEMAHIRAN BERFIKIR ALGEBRA DALAM PENYELESAIAN MASALAH BAGI PERSAMAAN LINEAR TINGKATAN SATU

The Algebraic Thinking Test (ATT) Instrument was developed to measure algebraic thinking skills in solving problems involving the topic of linear equations. This study meant to determine the quality of ATT instruments through the validity and reliability analysis using the Rasch Measurement Model. ATT comprises of nine closed ended questions with three constructs of algebraic thinking namely: Generalized Arithmetic (GA), Functions (F) and Modeling (M). The sample consisted of 120 students aged 12 to 13 years from one rural area school in Tuaran district, Sabah. Validity analysis found that the polarity of the items through the PTMEA-Corr value indicated that all ten items were >0.00 (+). Through item suitability analysis, all items are still retained as they meet the requirements in the range of one of the MNSQ, Outfit ZSTD and PTMEA-Corr outfit. In the Principal Components Analysis (PCA), the Raw Variance Explained by Measures was found 61.1% (excellent) and the Unexplained variance value in the first contrast below 15% indicate that the instrument has a strong dimensionality and has high construct validity. The reliability of Cronbach Alpha (KR-20) demonstrated a value of 0.90 (very high), item reliability of 0.98 (excellent) and respondent 0.86 (good) with an isolation value of item 6.29 and respondent 2.45. In conclusion, ATT has good validity and high reliability in measuring algebraic thinking skills among form one students in secondary schools.

Algebraic Thinking in the Context of Spatial Visualization

This pattern-related problem, appropriate for high school students, involves spatial visualization, promotes geometric and algebraic thinking, and relies on a no-cost computer software program.

Understanding the complexities of mathematical cognition: A multi-level framework.

Mathematics skills are associated with future employment, well-being, and quality of life. However, many adults and children fail to learn the mathematics skills they require. To improve this situation, we need to have a better understanding of the processes of learning and performing mathematics. Over the past two decades, there has been a substantial growth in psychological research focusing on mathematics. However, to make further progress, we need to pay greater attention to the nature of, and multiple elements involved in, mathematical cognition. Mathematics is not a single construct; rather, overall mathematics achievement is comprised of proficiency with specific components of mathematics (e.g., number fact knowledge, algebraic thinking), which in turn recruit basic mathematical processes (e.g., magnitude comparison, pattern recognition). General cognitive skills and different learning experiences influence the development of each component of mathematics as well as the links between them. Here, I propose and provide evidence for a framework that structures how these components of mathematics fit together. This framework allows us to make sense of the proliferation of empirical findings concerning influences on mathematical cognition and can guide the questions we ask, identifying where we are missing both research evidence and models of specific mechanisms.

Peningkatan Kemampuan Algebraic Thinking Siswa melalui Pendekatan Contextual Teaching and Learning (CTL)

The algebraic thinking ability of junior high school students remains low. This condition is indicated by their limited understanding of story questions. Students lack practice in expressing what is known and asked in story problems. One of the causes of this low algebraic thinking ability is the use of inappropriate learning models or approaches. This research aims to apply the Contextual Teaching and Learning (CTL) approach to enhance students' algebraic thinking skills. The purpose of this study is to determine whether the Contextual Teaching and Learning (CTL) approach can improve students' algebraic thinking skills. The population of this study consists of 77 students from class VIII of MTsN 6 Kerinci. The research sample includes 38 students divided into two groups. The experimental research method is employed in this study. The results indicate that the N-Gain in the experimental group falls within the moderate range, namely 0.55, while the control group's N-Gain falls within the low range, specifically 0.26. The results of the hypothesis test demonstrate that the Contextual Teaching and Learning (CTL) approach significantly improves students' algebraic thinking skills. Hence, it is recommended to implement the Contextual Teaching and Learning (CTL) approach in current instruction to support the enhancement of students' algebraic thinking skills.

Professional Learning Opportunities for Teachers of the Early Years in Algebra Teaching: A Study on the Practice of a Teacher Educator

Background: Teacher education processes directly influence the quality of teaching and, indirectly, students’ learning, which leads teacher educators to assume a central role in the development and leading of these processes, justifying in-depth studies about their actions. Objective: To understand how the actions and practices developed by a teacher educator during the orchestration of whole-group discussions contribute to creating professional learning opportunities about mathematical knowledge and educational practice regarding working with algebraic thinking aimed at elementary school teachers. Design: This is a qualitative-interpretive study along the lines of intervention research. Settings and Participants: The study included the teacher educator and 14 teachers who taught in grades 3-5 in a teaching network in a municipality in the state of São Paulo, Brazil. Data collection: The data were collected through video recording during the whole-group discussions with two cameras, one directed at the teacher educator and the other at the teachers. Results: The results indicate that the teacher educator used a set of actions and practices to conduct the whole-group discussions. Conclusions: The teacher educator’s practices that led teachers to reflect on their practice are the ones that most provide professional learning opportunities, even though other practices are essential in creating an environment that promotes whole-group discussions.

AN ALGEBRAIC THINKING SKILL TEST IN PROBLEM-SOLVING FOR SEVENTH GRADERS

Diagnostic tests have been developed previously to measure algebraic thinking skills; however, the tests do not specifically address algebraic problem-solving. Thus, an Algebraic Thinking Test (ATT) Instrument was developed to measure algebraic thinking skills in problem-solving involving linear equations. ATT comprises nine open-ended questions with three algebraic thinking constructs: Generalized Arithmetic, Functions and Modeling. Generalized arithmetic involves students in efficient calculation and generalization; functional involves identifying number patterns, while modelling involves solving open-ended problems, identifying similarities, and performing calculations involving variables. This study is meant to determine the quality of ATT instruments through the validity and reliability analysis using the Rasch Measurement Model. The sample consisted of 120 seventh graders aged 12 to 13, selected from two secondary schools in the Tuaran district, Malaysia. The instrument was found to have a strong dimensionality and high construct validity. The reliability of Cronbach Alpha (KR-20) demonstrated a value of .90 (very high), and item and respondent reliability of .98 (excellent) and .86 (good), respectively, with an item separation index of 6.29 and 2.45 for the person separation index. ATT has good validity and high reliability in measuring algebraic thinking skills among seventh graders in secondary schools. Keywords: algebraic thinking skills, linear equations, problem-solving, Rasch analysis, validity and reliability

우리나라 초등학교 수학에서 그림그래프 지도에 대한 고찰과 개선 방안 탐색

This study tried to analyze the problems by critically examining how the pictographs are taught for statistical literacy education in the Korean elementary school mathematics curriculum. First, I analyzed pictographs presented from the 4th mathematics curriculum to 2015 revised curriculum of Korea, in order to find the pedagogical goals of the pictographs in the elementary school mathematics textbooks. In addition, I derived some implications for teaching of pictographs by analyzing the shape of the pictographs in Australia, Singapore, and MiC mathematics textbooks. As a result, pictographs were taught for enhancing algebraic thinking in Korea by combining different sizes of pictures based on the power of 10 units. However, in other countries, the shape of pictographs were chosen based on the comprehension of the unit idea with the aim of developing the statistical literacy required to interpret the various pictographs found in daily life. For the teaching of pictographs for statistical literacy education, it is suggested to use the pictographs with only one size of pictures representing units and various units represented by pictures that are not powers of 10 in the elementary school mathematics curriculum.

The Use of ICT and the Realistic Mathematics Education on Learning Functions to Develop Algebraic Thinking Skills

The concept of function has evolved into one of the most fundamental principles in modern mathematics, influencing practically every area. However, it proved to be one of the most difficult ideas to grasp in mathematics. The concept of function is frequently taught without any connection to real-life situations. Students must work with problem scenarios that happen in a variety of contexts that are relevant to their everyday lives or make sense to them in order to gain authentic comprehension. This is consistent with the RME's properties. The unwillingness to highlight the visual element of the notion is one of the major roadblocks in learning functions. Using ICT and an RME approach to learning functions as an alternative to overcome these barriers could be a viable option. ICT and RME can help students learn mathematical principles in function content by providing a clearer visual experience. The purpose of this study is to investigate how ICT and RME can be used to improve learning function and increase algebraic thinking skills. This study conducted in a classroom of 29 junior high school students at the eighth grade. Design research methodology was used in this study by conducting three phases: preliminary design, teaching experiment, and retrospective analysis.

Crack the Code

This game teaches algebraic generalizations through differentiated play in pairs, small groups, or as a whole class and uses manipulatives to bridge numerical and algebraic thinking.

Examining the changes in mathematics teachers’ technology dispositions through GeoGebra-mediated professional development

Drawing on a technology dispositions framework, this study examined the changes in teachers’ beliefs, attitudes, competence, and behavioral intentions toward the use of technology in the mathematics classroom, following their engagement in professional development (PD) mediated by GeoGebra software. In a case study research design, 11 mathematics teachers from a senior high school in Ghana were purposively selected to engage in a 1-year PD program. They worked in teams to design and enact GeoGebra-based mathematics lessons. A self-report questionnaire, semi-structured interview, and lesson observation were used for data collection. Self-report data were analyzed using mean, SD, paired sample t-test, and Hedges’ g. Interview and lesson observation data were analyzed using a thematic approach. The results indicated that teachers’ pedagogical beliefs, affective attitudes, and competence toward technology changed following the PD. To some extent, they were able to make the goals of their mathematics lessons explicit and sequenced their instructional activities to engage students’ geometric and algebraic thinking. However, their intention to put the new pedagogy into classroom practice in the future depended on multiple contextual factors: administrative support, continual professional training, and provision of adequate technology facilities. These findings have implications for practices in mathematics education, technology dispositions frameworks, and PD.

Analysis of the Institutional Relationship of the Modeling Activity in a Moroccan High School Textbook

To identify the institutional relationship to modeling as an element of algebraic thinking, this article aims at analyzing the algebraic potential residing in the modeling activities that appear in the school textbook "Al-Moufid"; referring to the Anthropological Theory of Didactics (TAD) developed by Chevalard(1999) and to the Epistemological Reference Model of Algebraic Thinking (MERPA) proposed by Najar et al.( 2021). This official textbook is for 1st-year secondary Moroccan students (Age 12 to 13), as a level in which the arithmetic-algebra transition is manifested. In this institution, the pedagogical guidelines treat modeling as a skill to be developed in students without making an explicit definition or a link with the development of algebraic thinking. This analysis shows the considerable presence of modeling activities but with insufficient algebraic potential. On the other hand, several proposed modeling activities do not allow students to go completely through the modeling process as described by Chevallard (Chevallard, 1989).
 
 Received: 17 October 2022 / Accepted: 29 December 2022 / Published: 5 January 2023

The construction of algebraic thinking by deaf students from a systematic review of the literature

This article aims to analyze the specificities of the construction of algebraic thinking by deaf students in the early and final years of elementary school through a Systematic Review. The specific objectives are: to identify the curricular recommendations - National Common Curricular Base (2017) and Pernambuco Curriculum (2019) related to the construction of algebraic thinking by elementary school students; to retrieve from the CAPES Periodicals Portal, Brazilian Digital Library of Theses (BDTD) and Google Scholar research that deals with the construction of algebraic thinking or the learning of mathematical knowledge by deaf students; to point out the recommendations of the literature on the development of algebraic thinking by deaf elementary school students. The study was guided by a systematic literature review methodology based on the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA), which started with searches for articles and dissertations in the CAPES Periodicals Portal, BDTD and Google Scholar databases. A total of 2 727 studies were identified. After applying the inclusion or exclusion criteria, nine studies were selected for review. The results indicated that deaf students, like hearing students, present particularities in the construction of mathematical knowledge, especially in the Algebraic Unit Theme.

Professional Development of Mathematics Teachers’ Pedagogical Content Knowledge in Teaching Basic Algebra

This article presents the findings of the mathematics teachers’ professional development (PD) in teaching algebra from a continuously implemented two months PD program. The program was conducted as a case study for a group of 20 secondary level mathematics teachers who are following a two-year in-service teacher training course. This program targeted to improve the secondary level mathematics teachers’ pedagogical content knowledge (PCK) in teaching algebra. The content of the PD program included the nature of algebraic concepts, psychological aspects related to PCK, algebraic thinking and the algebraic learning styles. The result of the paired t-test indicated 0.345, a positive correlation between the post-test marks and pre-test marks. It interprets a favourable progress of the professional development (PD) program. From the interview findings, teacher trainees asserted the importance of algebraic thinking and the psychological foundation for making mathematics education effective and quality. The results asserted that the mathematics teachers’ PCK in algebra could be developed toa considerable extent from this PD program. The findings further revealed that mathematics teachers could develop their algebraic thinking from the program and it isa must. Since mathematics teachers’ PCK directly influence on students’ algebra learning and mathematics achievement, the findings of the study deepens the understanding that the mathematics teachers PCK must be developed to increase the students’ mathematics learning and understanding of the concepts. Therefore, I recommend that teacher education courses and continuous education programs must be developed in the teacher education curriculum with reforms for enhancing the quality of mathematics education in Sri Lanka.

The Generalization as a Tool to Develop the Algebraic Thinking Process

In this paper, we propose a reflection on how pupils use algebraic generalization and symbolization during the transition from a geometric thinking process to the algebraic one. This paper describes a new method/idea based on the use of geometric patterns in some proposed learning situations with the primary aim of understanding how this use may enable learners to develop and improve their algebraic thinking process. Thus, we proposed to a sample of pupils from 10 to12 years old to draw some geometric figures of squares and rectangles and asked them to conjecture the three well-known remarkable identities, (a+b) 2 , (a-b) 2 and a 2 -b 2 . The students are supposed to put these squares and rectangles together to form a single square or a rectangle and to deduce these identities. Our analysis focuses on videos capturing student behavior using patterns to develop some generalization of the algebraic process on data collected during student interviews and on the difficulties of students deriving algebraic formulas from some geometric patterns. Keywords: generalization, algebraic thinking process, geometric thinking process, patterns in education. https://doi.org/10.55463/issn.1674-2974.50.2.16

The intertwining of the students’ Algebraic Thinking development and the teacher’s professional constitution: revelations of the pedagogical narrative of a teacher-researcher

Resumo Este artigo tem como objetivo discutir processos do desenvolvimento do Pensamento Algébrico de alunos do primeiro ano do Ensino Fundamental entrelaçado aos processos de constituição profissional da professora-pesquisadora, que tomou como foco de estudo a pesquisa na/da própria prática e as aprendizagens docente e discentes. A presente discussão faz parte de uma pesquisa de doutorado, de abordagem qualitativa, com apoio financeiro da Capes[1]. O cenário de investigação pertencia à rede municipal de uma cidade do interior de São Paulo, uma sala de aula composta por 26 alunos. As reflexões teóricas pautam-se na perspectiva histórico-cultural, nos estudos sobre Pensamento Algébrico e nas narrativas pedagógicas como dispositivo de autoformação e desenvolvimento profissional. Para a análise, apresenta-se uma narrativa pedagógica sobre uma tarefa que aborda uma sequência com padrão recursivo e objetiva a percepção da regularidade e o desenvolvimento de estratégias de generalização. A escrita e a análise narrativa, no contexto da pesquisa na/da própria prática, revelam indícios da constituição profissional da professora, bem como mostram que os alunos do primeiro ano do Ensino Fundamental, engajados numa cultura social de sala de aula de Matemática pautada na problematização, foram capazes de generalizar o padrão de sequências por meio da linguagem materna.

Approche par problème et formation d’enseignants de mathématiques : comment se diffusent, en formation, les résultats de la recherche ?

Based on the framework of meta-didactic transposition analysis (Arzarello et al., 2014) and specifically the concepts of brokering and boundary object, this paper studies how knowledge to teach algebra is exchanged during a training program involving nine mathematics teachers and two researchers specialised in algebra teaching and learning. Organized in three half-day sessions, this program is based on a problem pointed out in the research literature as particularly rich to develop algebraic thinking. In addition, the materials used in training are come directly from the classes of the teachers participating in the program. In this sense, the program values knowledge that makes sense in both research and teaching practice. The analysis of interactions between researchers and teachers highlights three types of collaborative activities between the two groups and thus questions the potential of such a mechanism to foster integration of research results by teachers.

Investigating the Mathematics Teachers’ Pedagogical Content Knowledge in Teaching Algebraic Symbols

Secondary level mathematics teachers’ pedagogical content knowledge (PCK) in teaching algebra is identified as a crucial factor of mathematics teachers’ knowledge that influences on the accuracy of the students’ understanding. The aim of this study is to examine the extent of the mathematics teachers’ PCK in teaching basic algebraic concepts, algebraic symbols. Necessity of learning algebra for advanced conceptualization is asserted by the global literature. Researchers found that the mathematics teachers’ PCK in teaching algebra is not satisfactory. Equally, students’ learning difficulties and their struggle in learning algebra can be identified as a global issue. The sample (N=281) mathematics teachers in the Galle education zone were administered with a questionnaire. The mean value was 6.81 and the standard deviation was 2.74. The study found that the mathematics teachers’ PCK in teaching algebraic symbols is very poor in Sri Lanka. Moreover, the mathematics teachers use inadequate algebraic thinking in the teaching practicum that can restrict the students’ creativity. Students’ understanding in algebraic symbols at the acquisition stage of learning algebra should essentially be developed through well designed professional development programs.

On study the fractional Caputo-Fabrizio integro differential equation including the fractional q-integral of the Riemann-Liouville type

<abstract><p>The major objective of this scheme is to investigate both the existence and the uniqueness of a solution to an integro-differential equation of the second order that contains the Caputo-Fabrizio fractional derivative and integral, as well as the q-integral of the Riemann-Liouville type. The equation in question is known as the integro-differential equation of the Caputo-Fabrizio fractional derivative and integral. This equation has not been studied before and has great importance in life applications. An investigation is being done into the solution's continued reliance. The Schauder fixed-point theorem is what is used to demonstrate that there is a solution to the equation that is being looked at. In addition, we are able to derive a numerical solution to the problem that has been stated by combining the Simpson's approach with the cubic-b spline method and the finite difference method with the trapezoidal method. We will be making use of the definitions of the fractional derivative and integral provided by Caputo-Fabrizio, as well as the definition of the q-integral of the Riemann-Liouville type. The integral portion of the problem will be handled using trapezoidal and Simpson's methods, while the derivative portion will be solved using cubic-b spline and finite difference methods. After that, the issue will be recast as a series of equations requiring algebraic thinking. By working through this problem together, we are able to find the answer. In conclusion, we present two numerical examples and contrast the outcomes of those examples with the exact solutions to those problems.</p></abstract>

ALJABAR UNTUK ANAK USIA DINI

This paper aims to describe the importance of teaching algebraic content and algebraic thinking skills for early childhood and how the learning strategies are at the early childhood education level. The research method uses library research to understand learning algebraic content and algebraic thinking for early childhood. The results obtained in this study include the application of learning mathematics for children through various creative activities and introducing the concept of arithmetic and geometric concepts. Both of these concepts in teaching early childhood are introduced to several objects in the environment around them so that these mathematical concepts can be understood and understood more deeply by early childhood.

Comparison of the Mathematics Contents of Turkish and Swiss Preschool Curricula

This study aimed to compare the mathematics contents of Turkish and Swiss Preschool Curricula. In this context, the similarities and differences of the outcomes belonging to the learning domains were expressed. The research was designed qualitatively and document analysis was used as the data collection technique. The mathematics contents of the Turkish and Swiss Preschool Curricula were taken into consideration as documents. Content analysis was used as the data analysis method. According to the findings obtained in this study, it was determined that the preschool curricula in Turkey and Switzerland are organized according to developmental areas, the learning domains are similar, and both countries detail the curriculum with indicators depending on the stated outcomes. In both curriculum, the outcomes and indicators for the learning domains of "knowing numbers and using them in daily life, operations and algebraic thinking, recognizing and using patterns, recognizing basic geometric shapes, recognizing basic spatial concepts and using them in daily life, recognizing and using measurement units" were included. In the light of the results obtained depending on these achievements and indicators, it is only in the Swiss curriculum that the indicator of reading and writing numbers for the achievement of knowing numbers and using them in daily life is; It has been determined that Swiss is at a higher cognitive level in terms of operations and algebraic thinking. In the acquisition of recognizing and using patterns, the stage of "creating one's own pattern" is not addressed in the Swiss curriculum; It has been concluded that the Turkish curriculum is in a more positive approach for the acquisition of recognizing and using measurement units.