597 results on '"Early Algebra"'
Search Results
2. The professional practice of designing tasks: how do pre-service early childhood teachers promote mathematical processes in early algebra?
- Author
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Alsina, Ángel, Pincheira, Nataly, and Delgado-Rebolledo, Rosa
- Subjects
EARLY childhood teachers ,EARLY childhood education ,STUDENT teachers ,MATHEMATICS education ,INTEGRALS - Abstract
Spanish educational curriculum adopts a mathematical process-based approach, which encompasses problem solving, reasoning and proof, communication, connections and representation. A fundamental role in the integration of these processes in mathematics teaching is played by teachers' professional practice of designing tasks. According to this, our aim is to analyze the ways in which pre-service early childhood teachers understand the mathematical processes in the professional practice of designing early algebra tasks and to identify how they intend to promote these processes through the tasks. Content analysis techniques were used to examine the designed tasks. To illustrate the data analysis and results, six tasks are presented. As a result, pre-service early childhood teachers associate problem solving with challenges and questions. They understand problems as unfamiliar situations but ignore the relationships between students and tasks. Moreover, they do not encourage exploration of phases of problem solving and tend to use strategies more suitable for routine tasks. Communication is identified in all the tasks designed, encouraging interaction and discussion. However, only one task explicitly promotes mathematical language. For reasoning and proof, pre-service teachers begin to use questions to elicit explanations and justifications, but do not encourage verification strategies and various modes of reasoning. The process of connections is only present in one task, reflecting the fragmented nature of mathematics teaching. We conclude that the professional practice of designing mathematical tasks is a powerful in teacher education. However, training programs should place greater emphasis on the meaningful use of mathematical processes. [ABSTRACT FROM AUTHOR] more...
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- 2024
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3. Semiotic mediation of gestures in the teaching of early algebra: the case of the equal sign.
- Author
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Stylianou, Despina A., Lee, Boram, Ristroph, Ingrid, Knuth, Eric, Blanton, Maria, Stephens, Ana, and Gardiner, Angela
- Subjects
- *
SEMIOTICS , *GESTURE , *ALGEBRA , *MATHEMATICS education , *COGNITION - Abstract
Gestures are one of the ways in which mathematical cognition is embodied and have been elevated as a potentially important semiotic device in the teaching of mathematics. As such, a better understanding of gestures used during mathematics instruction (including frequency of use, types of gestures, how they are used, and the possible relationship between gestures and student performance) would inform mathematics education. We aim to understand teachers' gestures in the context of early algebra, particularly in the teaching of the equal sign. Our findings suggest that the equal sign is a relatively rich environment for gestures, which are used in a variety of ways. Participating teachers used gestures frequently to support their teaching about the equal sign. Furthermore, the use of gestures varied depending on the particular conception of the equal sign the instruction aimed to promote. Finally, teacher gesture use in this context is correlated with students' high performance on an early algebra assessment. [ABSTRACT FROM AUTHOR] more...
- Published
- 2024
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4. How tools mediate elementary students' algebraic reasoning about evens and odds.
- Author
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Strachota, Susanne, Stephens, Ana, Morton, Karisma, Veltri-Torres, Ranza, Blanton, Maria, Gardiner, Angela Murphy, Sung, Yewon, Stroud, Rena, and Knuth, Eric
- Subjects
ODD numbers ,GRADING of students ,STUDENTS - Abstract
This study investigated the role of tools in supporting students to reason about even and odd numbers. Participants included Kindergarten, Grade 1, and Grade 2 students (ages 5–8) at two schools in the USA. Students took part in a cross-sectional early algebra intervention in which they were asked to generalize, represent, justify, and reason with mathematical structure and relationships. Interviews were conducted with students before, during, and after the year-long intervention to explore the ways in which tools mediated their engagement in these practices in the context of questions about parity. In addition to cubes and "dot cards," which were introduced during the interviews, we found that students also used their fingers and hands as tools. We outline the affordances and constraints of these tools in supporting students' reasoning about the properties of even and odd numbers. [ABSTRACT FROM AUTHOR] more...
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- 2024
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5. For a New Approach of the Quantitative Equivalence
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Rinaldi, Anne-Marie, Florensa, Ignasi, editor, Ruiz-Munzón, Noemí, editor, Markulin, Kristina, editor, Barquero, Berta, editor, Bosch, Marianna, editor, and Chevallard, Yves, editor
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- 2024
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6. Due esperienze didattiche di early-algebra con il DIST-M: dalla costruzione del problema all’implementazione in aula
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Maria Aceto, Maria Lodina De Santis, Angela Donatiello, Giuseppina Lemmi, Simona Manzoni, Concetta Rosaria Montervino, Luca Picariello, Valeria Scaramuzzino, and Giovanna Vadalà
- Subjects
argomentazione ,pensiero relazionale ,problemi-storia ,rappresentazioni semiotiche ,early algebra ,Education ,Mathematics ,QA1-939 - Abstract
La possibilità di approcciare al discorso algebrico mediante attività che mettano in luce diverse rappresentazioni permette agli studenti lo sviluppo del pensiero relazionale. Le esperienze presentate sono relative alla progettazione di attività a carattere fortemente immersivo che hanno coinvolto studenti di scuole secondarie di primo grado nello sviluppo della competenza argomentativa e del formalismo simbolico in ambito di teoria dei numeri. L’approccio narrativo e immersivo è stato possibile grazie all’uso del dispositivo didattico del DIST-M, mentre l’utilizzo di sistemi semiotici visuali ha favorito la formazione di un concetto in una prospettiva vygotskijana. more...
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- 2024
7. Assessing knowledge to teach early algebra from the Mathematical Knowledge for Teaching (MKT) perspective: A support tool for primary school teachers
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Nataly Pincheira and Ángel Alsina
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assessment instrument ,early algebra ,mathematical knowledge for teaching ,pre-service teachers ,primary education ,Mathematics ,QA1-939 - Abstract
Research about the optimal methodologies for guiding the training of primary school instructors in the instruction of early algebra remains in the process of determination. This manuscript delineates the methodology involved in developing and validating an assessment tool to evaluate the mathematical acumen of prospective primary educators in the domain of early algebra during their initial pedagogical training, drawing upon the constructs delineated in the Mathematical Knowledge for Teaching (MKT) model. To this end, an instrumental inquiry comprising four distinct phases has been executed: an exhaustive review of extant literature concerning the mathematical proficiency of primary school instructors and the pedagogy of early algebra; formulation of the preliminary version of the assessment instrument; validation of said instrument via expert appraisal and a preliminary application involving ten pre-service primary educators enrolled in a Spanish academic institution; and subsequent refinement and finalization of the assessment tool. This endeavor's outcome culminated in the MKT-Early Algebra Questionnaire (6-12-year-olds), comprising six open-ended inquiries meticulously designed to explore diverse facets of early algebra while aligning with the various sub-domains delineated in the MKT model. It is deduced that the resultant instrument holds promise as an effective diagnostic apparatus, serving dual purposes: elucidating the mathematical proficiency of primary school educators in the context of early algebra and fostering introspection regarding pedagogical strategies conducive to the cultivation of algebraic cognition at this developmental juncture. By furnishing a comprehensive questionnaire that systematically addresses all facets of pedagogical knowledge requisite for the effective instruction of early algebra, this study furnishes invaluable insights into the specific components that should be integrated into teacher education curricula in the domain of algebraic didactics. more...
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- 2024
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8. Representations of Numerical Sequences in Mexican Secondary School Textbooks.
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Hernández, Yasmin Cruz, Juárez López, José Antonio, Avelli, Carolina Napp, and García, Lizzet Morales
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MATHEMATICAL sequences ,TEXTBOOKS ,SECONDARY schools ,MATHEMATICAL analysis ,MATHEMATICS textbooks - Abstract
The study of numerical sequences usually focuses on the use of symbols or finding generalizations, that is, a more formal notation. However, there is several research on the different representations, but little research in representations that we consider as drawings and illustrations and that, in general, are commonly used in textbooks when both a numerical sequence and a figurative sequence are shown. The present research with a qualitative approach sought to identify the representations of sequences contained in Middle school textbooks. Nine textbooks were analyzed using the theoretical-methodological proposal known as Mathematical Content Analysis, where 28 different figures were found to represent a numerical sequence. The findings showed that there are three types of representations in the textbooks analyzed: abstract figures, real figures, and pictorial figures. The results show that the representations that appear most frequently in textbooks are pictorial figures. [ABSTRACT FROM AUTHOR] more...
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- 2024
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9. Generalizing actions with the subtraction-compensation property: primary students' algebraic thinking with tasks involving vertical towers of blocks.
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Wilkie, Karina J. and Hopkins, Sarah
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- *
ARITHMETIC , *VISUALIZATION , *PENSIONS , *MATHEMATICS , *ALGEBRA - Abstract
An important approach for developing children's algebraic thinking involves introducing them to generalized arithmetic at the time they are learning arithmetic. Our aim in this study was to investigate children's attention to and expression of generality with the subtraction-compensation property, as evidence of a type of algebraic thinking known as relational thinking. The tasks involved subtraction modelled as difference and comparing the heights of towers of blocks. In an exploratory qualitative study, 22 middle primary (9–11-year-old) students from two schools participated in individual videoed interviews. The tasks were designed using theoretical perspectives on embodied visualization and concreteness fading to provide multiple opportunities for the students to make sense of subtraction as difference and to advance their relational thinking. Twelve out of 22 students evidenced conceptual understanding of the comparison model of subtraction (subtraction as difference) and expression of the compensation property of equality. Four of these students repeatedly evidenced relational thinking for true/false tasks and open equivalence tasks. A proposed framework for levels of attention to/expression of generality with the subtraction-compensation property is shared and suggestions for further research are presented. [ABSTRACT FROM AUTHOR] more...
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- 2024
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10. Research trends on early algebra in the middle school: A combined bibliometric and meta-analysis review
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Dilham Fardian, Didi Suryadi, Sufyani Prabawanto, and Silfia Hayuningrat
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comprehensive meta analysis ,bibliometric analysis ,early algebra ,meta analysis ,middle school ,Rstudio ,Mathematics ,QA1-939 - Abstract
The aims of this study is to analyze publications on early algebra in middle schools to contribute to the development of related literature. A total of 234 articles on early algebra published in various Scopus databases between 2013 and 2024 were retrieved and analyzed through a bibliometric analysis approach and 19 articles through a Comprehensive Meta Analysis (CMA). The results showed fluctuating trends in research related to early algebra in middle schools. The surge in publications using VOSviewer and RStudio from 2013 to 2024 did not align with a corresponding increase in average citations, contributing to a stagnation in the average citation per year. The noticeable upward trend in the volume of publications on early algebra since the beginning of 2013 indicates a dynamic and evolving research landscape in this field. This consistency suggests a widespread and sustained interest in early algebra research beyond the confines of a single country, further emphasizing its global relevance and significance. Integrating multimedia technology into early algebra instruction significantly enhances student learning outcomes. Moreover, access to technology and enhanced learning resources through technology integration can play a more crucial role in improving the effectiveness of early algebra learning in countries with lower income levels. more...
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- 2024
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11. Theories with semiotic approaches: possibility of an intertheoretical dialogue
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Renata Aparecida de Faria
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early algebra ,semiótica ,teoria da objetivação ,teoria dos registros de representação semiótica. ,Special aspects of education ,LC8-6691 ,Mathematics ,QA1-939 - Abstract
The theories presented in this text have a semiotic approach, but different focuses: the Theory of Semiotic Representation Records (TRRS), developed by Raymond Duval, presents a cognitive approach in which representation records are inherent to the process of teaching and learning mathematics and enable the analysis of cognitive, treatment and conversion activities. The socio-cultural approach that delimits the assumptions of the Theory of Objectification (TO), a theory in development proposed by Luis Radford, considers Common Labor as fundamental to the processes of objectification and subjectification, anchored by semiotic means of objectification. In this way, we indicate the specificities regarding the principles, methodology and investigative questions of each theory followed by resolutions of a situation in the context of Early Algebra, with 6th year students from a public school in the north of Paraná, as an illustration for analysis under the lenses of the TO and TRRS framework. The possibility of an intertheoretical dialogue arises from the specificities and inferences of the analysis regarding the elements of each theory, where it is concluded that even with distinct ontological and epistemological aspects, both theories emerge synchronously more...
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- 2023
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12. Progressions in young learners' understandings of parity arguments.
- Author
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Blanton, Maria, Gardiner, Angela Murphy, Ristroph, Ingrid, Stephens, Ana, Knuth, Eric, and Stroud, Rena
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- *
ARGUMENT , *GRADING of students - Abstract
Understanding how young learners come to construct viable mathematical arguments about general claims is a critical objective in early algebra research. The qualitative study reported here characterizes empirically developed progressions in Grades K–1 students' thinking about parity arguments for sums of evens and odds, as well as underlying concepts of pair and parity of a number. Data are drawn from classroom lessons of a Grades K–1 early algebra instructional sequence, as well as task-based interviews conducted at four timepoints during the implementation of the sequence. While most students at the beginning of the study (Grade K) did not know the concepts of even or odd and could not make any viable arguments regarding parity, by the end of Grade 1 students were largely constructing representation-based arguments to justify number parity and claims about sums of evens and odds. Results of this study align with other research that shows young learners can develop viable arguments to justify mathematical generalizations. Results also provide preliminary evidence that the instructional sequence used here can foster students' practice of argumentation from the start of formal schooling. [ABSTRACT FROM AUTHOR] more...
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- 2024
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13. Generalization strategies and representations used by final-year elementary school students.
- Author
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Ureña, Jason, Ramírez-Uclés, Rafael, Cañadas, María C., and Molina, Marta
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GENERALIZATION , *ELEMENTARY schools , *ALGEBRA , *REASONING , *DEPENDENT variables - Abstract
Recent research has highlighted the role of functional relationships in introducing elementary school students to algebraic thinking. This functional approach is here considered to study essential components of algebraic thinking such as generalization and its representation, as well as the strategies used by students and their connection with generalization. This paper jointly describes the strategies and representations of generalization used by a group of 33 sixth-year elementary school students, with no former algebraic training, in two generalization tasks involving a functional relationship. The strategies applied by the students differed depending on whether they were working on specific or general cases. To answer questions on near specific cases they resorted to counting or additive operational strategies. As higher values or indeterminate quantities were considered, the strategies diversified. The correspondence strategy was the most used and the common approach when students generalized. Students were able to generalize verbally as well as symbolically and varied their strategies flexibly when changing from specific to general cases, showing a clear preference for a functional approach in the latter. [ABSTRACT FROM AUTHOR] more...
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- 2024
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14. The Effects of a Functional Thinking Intervention on Fifth-Grade Students' Functional Thinking.
- Author
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AKIN, Gülnur and İŞLER BAYKAL, Işıl
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EXPERIMENTAL groups ,FUNCTIONAL groups ,MIDDLE schools ,QUANTITATIVE research ,STUDENTS - Abstract
This study aims to investigate the effects of a functional thinking intervention on 5th-grade students' functional thinking. Forty-three fifth-grade students from two public middle schools in Ankara participated in the study, in which 20 of them formed the experimental group in one school, and 23 of them formed the control group in the other school. The experimental group participated in a functional thinking intervention lasting 12 hours (about three weeks). A Functional Thinking Test (FTT) was administered to both groups as a preand post-test. The qualitative analysis of students' functional thinking strategies supported quantitative analysis. The results revealed no significant mean difference between the experimental and control groups at the pre-test or post-test. However, the experimental group showed significant pre-to-post gains. Also, experimental group students were significantly better at using variables in defining the function rule after the functional thinking intervention. [ABSTRACT FROM AUTHOR] more...
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- 2024
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15. The Effects of an Early Algebra Intervention on Third-Grade Students' Algebraic Thinking Skills.
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İŞLER-BAYKAL, Işıl, ÖZTÜRK-TAVŞAN, Nejla, GÜZELLER, Gizem, and SAYGILI, İlkay
- Subjects
- *
ALGEBRA , *GENDER inequality , *MENTAL arithmetic , *CONTROL groups , *STUDENTS - Abstract
The importance of early algebra has been emphasized in international literature, and it has been discussed in many studies that students who are involved in a comprehensive intervention improve their algebraic thinking skills. The aim of this study was to test the effects of an early algebra intervention on the algebraic thinking skills of 3rd-grade students. The 3rd-grade intervention and control groups were included in the study, and both groups were given pre-test and post-test. In the findings, although there was no statistical difference between the 3rd-grade intervention and control group performances in the pre-test, a statistically significant difference was found in the post-test. Analysis of the students' strategies revealed that the students in the intervention group used algebraic thinking strategies more in the post-test than the control group in three big ideas, which are equality, expressions, equations and inequalities, generalized arithmetic, and functional thinking. [ABSTRACT FROM AUTHOR] more...
- Published
- 2023
- Full Text
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16. The Developmental Progression of Early Algebraic Thinking of Elementary School Students.
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Sun, Siyu, Sun, Dandan, and Xu, Tianshu
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- *
SCHOOL children , *COGNITIVE development - Abstract
Developing algebraic thinking in elementary school has gained consensus among mathematics educators. The objective of this study is to understand the developmental trajectory of early algebraic thinking in elementary school students so as to assist teachers and curriculum developers in implementing instruction that aligns with students' cognitive development. This study adopted a cross-sectional survey approach, involving 526 students from grades three to five in Shanghai, who were tested using a 12-item assessment that measured three aspects: "generalized arithmetic", "functional thinking", and "quantitative reasoning". Latent class analysis was used to analyze students' response strategies, and, in conjunction with individual interviews, this study identified potential developmental pathways in students' early algebraic thinking, progressing from "arithmetic thinking" to "concrete algebraic thinking", "generalized algebraic thinking", and finally to "symbolic algebraic thinking". As thinking levels advanced, significant differences in students' response strategies emerged, with notable improvements in "generalization abilities" and "symbolization abilities". This study suggests that educational practices should encompass content in elementary arithmetic curricula that fosters generalization abilities. Additionally, providing students with opportunities for diverse representations can effectively stimulate the development of early algebraic thinking. [ABSTRACT FROM AUTHOR] more...
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- 2023
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17. Strategies and representations used by early childhood education students in a functional thinking task: A case study.
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Morales, Rodolfo, Pizarro, Fernanda, Díaz-Levicoy, Danilo, and García-García, Jaime I.
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EARLY childhood education ,ALGEBRA ,GENERALIZATION ,TASK performance ,ACQUISITION of data - Abstract
This article presents a study that addressed the functional relationships that two early childhood education students (five-six years old) evidenced, as well as the representations they used when solving a functional thinking task. The proposed task involved the function f(x)=2x, with six questions on particular cases and one on generalization. The data was collected through a semi-structured interview to each of the students and a qualitative analysis of their answers was carried out in each of the questions of the task. The results suggest that the two students are capable of approaching the proposed task through different strategies, such as additive and multiplicative correspondence relationship, and covariation. Also, it was found that they use systems of varied representations, being the verbal representation to express the generalization of the functional relationship one that stands out. It is concluded that early childhood education students may be able to tackle tasks that involve algebraic notions that focus on functional thinking. [ABSTRACT FROM AUTHOR] more...
- Published
- 2023
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18. Development of prospective elementary teachers’ knowledge to teach early algebra through case discussions
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Öztürk-Tavşan, Nejla and İşler-Baykal, Işıl
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- 2024
- Full Text
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19. On Teaching of Word Problems in the Context of Early Algebra
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Malara, Nicolina A., Telloni, Agnese I., Toh, Tin Lam, editor, Santos-Trigo, Manuel, editor, Chua, Puay Huat, editor, Abdullah, Nor Azura, editor, and Zhang, Dan, editor
- Published
- 2023
- Full Text
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20. EFFECTIVE INSTRUCTIONAL PRACTICES THAT FOSTER THE DEVELOPMENT OF STUDENTS' EARLY ALGEBRAIC THINKING.
- Author
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Lee, Boram, Knuth, Eric, Stephens, Ana, Ristroph, Ingrid, Blanton, Maria, Stylianou, Despina, Gardiner, Angela M., and Stroud, Rena
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EDUCATIONAL psychology ,MATHEMATICS education ,STUDENT engagement ,ALGEBRA ,CLASSROOM management - Abstract
Is it possible to identify instructional practices that have an impact on student learning in mathematics? The study described here is part of ongoing efforts to understand and characterize effective instruction. We drew on the work of several recently developed frameworks for understanding teaching effectiveness to develop a protocol for studying effective instruction that both coordinates and extends existing research in the context of early algebra. Using a largescale study, we characterized effective instruction in this context and documented the impact of such instruction on students' performance using both qualitative and quantitative analyses. Findings suggest that teachers' abilities to take up curriculum openings are important aspects of teaching. Furthermore, the manner with which teachers react to these moments strongly correlates with gains in student performance. [ABSTRACT FROM AUTHOR] more...
- Published
- 2023
21. EXPLORING LEARNING OPPORTUNITIES FOR PRIMARY TEACHERS: THE CASE OF KNOWLEDGE FOR TEACHING EARLY ALGEBRA.
- Author
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JACQUES RIBEIRO, ALESSANDRO, AGUIAR, MARCIA, TREVISAN, ANDRÉ LUIS, and RIZEK ELIAS, HENRIQUE
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- *
TEACHER development , *PROFESSIONAL employee training , *IN-service training of teachers , *TEACHER educators , *EVIDENCE gaps - Abstract
Understanding how to constitute and develop opportunities for primary teachers teach early algebra to younger children is still an important research gap. In this paper, we bring results of a research program developed in Brazil over the past five years. We aim to discuss how professional learning opportunities emerged when teachers collectively planned, discussed, and analyzed lessons involving different meanings of the equality symbol and the development of functional thinking. Developed from the perspective of a qualitative-interpretative research, data analyzed consists of curriculum documents, protocols for the resolution of formative tasks, audios and videos collected during teacher education processes for in-service teachers. The results highlight that professional learning tasks, combined with the actions of teacher educators during collective discussions, favored in-service teachers to differentiate and understand the students' reasoning. Some implications for teacher education as well as the professional development of primary teachers are discussed, especially regarding early algebra thinking, because teachers do not normally have the opportunity to study these contents on their own experiences in school. [ABSTRACT FROM AUTHOR] more...
- Published
- 2023
- Full Text
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22. THE ALGEBRAIC REASONING AND BLENDED FORMATION OF TEACHERS WHO TEACH MATHEMATICS: THE POWER OF SYMBOLS
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Caio Oliveira and Sandra Magina
- Subjects
early algebra ,símbolo ,formação continuada de professores ,intervenção de ensino ,ensino híbrido ,Education ,Special aspects of education ,LC8-6691 - Abstract
This article aims to analyze tasks developed by multipurpose teachers within the scope of a study that involved continuing education, to work on elementary algebraic concepts. The study is located at the intersection of Early Algebra, Blended Learning and Teacher Education. A continuing education course was developed for nine teachers who teach Mathematics, in the Elementary School, enrolled in a discipline of a Master's course in Education at a public university in the state of Bahia. For the purpose of this article, the discussion focused on a fundamental concept for the study of Algebra: the symbol. Data were produced in two environments (AVA REPARE and classroom) and consisted of: forums, discussion in the elaboration of problem situations and expositions of situations. The results point to the reflections carried out by the course-teachers, which led to a (trans)formation in their pedagogical practices. The interactivity of AVA REPARE provided a plurality of data, exchanges of experiences and rich information among the course-professors. Also, the study shows that the course carried out proved to be a rich and innovative model for in-service training of teachers who work at this level of education. more...
- Published
- 2023
- Full Text
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23. Factors influencing students' understanding of mathematical equivalence
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Simsek, Emine
- Subjects
510.1 ,mathematical equivalence ,equals sign ,cross-cultural ,early algebra ,primary mathematics - Abstract
Most primary school students have difficulty understanding mathematical equivalence, and the literature has shown that the situation is worse in some countries than others. Differences between students' understanding have significant and long-lasting effects, with understanding of equivalence predicting arithmetic and algebra achievement throughout school years. Currently, little is known about the factors influencing students' understanding of mathematical equivalence. This thesis explores the individual and classroom-level factors that influence students' understanding of mathematical equivalence and discusses the implications of the findings for future research, classroom practice, and policy. The first study investigated the role of the substitution conception of the equals sign on algebra performance and found that endorsement of substitution predicts secondary school students' algebra performance. In the second, the findings of the first were not replicated with undergraduate students. Next, the third study investigated whether problem features influence adults' equation-solving performance, and found that use of shortcut and problem size related to adults' performance. My final study was the first large-scale cross-cultural study in the field researching the individual and classroom level factors influencing primary students' understanding of equivalence. Moreover, I used advanced statistical analysis methods, i.e. multilevel structural equation modelling. The findings of this study showed that (i) students' knowledge of definitions of the equals sign, and (ii) teachers' knowledge of students' relational strategies related to students' understanding of equivalence, but (iii) teachers' knowledge of students' operational strategies and (iv) textbooks did not. The findings also showed that these relationships were similar across the participating countries (namely, China, England, New Zealand, South Korea, Turkey and the US). The results from the studies I conducted within the scope of this thesis make an original contribution to understanding factors influencing students' understanding of mathematical equivalence, and have theoretical, methodological, and practical implications. Implications discussed in the thesis could inform practitioners and policy makers in particular about where to invest for improving students' understanding of mathematical equivalence. more...
- Published
- 2020
- Full Text
- View/download PDF
24. Young Students' Arithmetic-Algebraic Structure Sense: an Empirical Model and Profiles of Students.
- Author
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Pittalis, Marios
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PATTERNS (Mathematics) ,TRACE analysis ,MENTAL arithmetic ,FACTOR structure ,STRUCTURAL models ,SENSES ,ARITHMETIC - Abstract
A theoretical model describing young students' (Grade 3) arithmetic-algebraic structure sense was formulated and validated empirically (n = 130), hypothesizing that young students' arithmetic-algebraic structure sense consists of five distinct but correlated factors; structure in numerical equivalence and word-problem modeling, structure in arithmetic, mathematical structure in number sequences, varying quantities and number correspondences, structure in patterns, and structure in functions. Data analysis suggested that arithmetic and arithmetic-algebraic tasks can be categorized based on the proposed model. Analysis traced three categories of students that represent different profiles of students. Students of the basic arithmetic structure sense profile approach flexibly simple arithmetic and patterning tasks that do not require generalization; students of the advanced arithmetic structure profile exhibit an awareness of local relations by describing the underlined relations in a variety of numerical situations. Students of the emergent arithmetic-algebraic profile utilize the awareness of mathematical structural relations to symbolize functional relations. A structural model showed the importance of grasping structure in patterns and describing the mathematical structure in numerical situations to develop structure in functions. [ABSTRACT FROM AUTHOR] more...
- Published
- 2023
- Full Text
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25. Transformando el conocimiento para enseñar matemáticas de docentes en formación de educación infantil a través del diseño de tareas.
- Author
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ALSINA, Ángel, PINCHEIRA, Nataly, and DELGADO-REBOLLEDO, Rosa
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EARLY childhood education ,EARLY childhood teachers ,STUDENT teachers ,ALGEBRA ,MATHEMATICS education ,MATHEMATICS ,RELATION algebras - Abstract
Copyright of Revista Interuniversitaria de Formación del Profesorado is the property of Asociacion Universitaria de Formacion del Profesorado (AUFOP) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) more...
- Published
- 2023
- Full Text
- View/download PDF
26. Developing and Reporting Psychometric Evidence of Prerequisite Algebra Skills Instrument.
- Author
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Kartika, Hendra, Budiarto, Mega Teguh, Fuad, Yusuf, Kim Jeonghyeon, and Bonyah, Ebenezer
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PSYCHOMETRICS ,SECONDARY school students ,MATHEMATICS education ,TEST validity ,GRADING of students ,MATHEMATICS teachers ,MATHEMATICS students ,ACQUISITION of data - Abstract
Copyright of Acta Scientiae (1517-4492) is the property of Acta Scientiae and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) more...
- Published
- 2023
- Full Text
- View/download PDF
27. STEM and Early Algebra: Reflections from Primary School Teachers' Practices.
- Author
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AYDOĞAN, Demet and BÜYÜKŞAHİN, Yasemin
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PRIMARY school teachers ,STEM education ,ALGEBRA education ,QUALITATIVE research ,CLASSROOM environment ,POSITIVE psychology - Abstract
Copyright of Instructional Technology & Lifelong Learning is the property of Instructional Technology & Lifelong Learning and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) more...
- Published
- 2023
- Full Text
- View/download PDF
28. Students' Learning Obstacles in Solving Early Algebra Problems: A Focus on Functional Thinking.
- Author
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Utami, Nadya Syifa, Prabawanto, Sufyani, and Suryadi, Didi
- Subjects
STUDENT development ,SCHOOL children ,ALGEBRA ,MATHEMATICAL variables ,TEACHING methods - Abstract
This study describes students' learning obstacles in solving early algebra problems requiring functional thinking ability. To reach this aim, qualitative research was conducted in this study. Participants of this study were 39 ninth graders and a mathematics teacher at one of the lower secondary schools in Bandung, Indonesia. The data were collected through the written test about early algebra problems, interviews, and document study. The findings revealed that fewer students achieve the correspondence level in their functional thinking ability. Many of them are on covariation or recursive patterns level. The variety of students' functional thinking levels in solving the problem is influenced by their previous learning experiences with early algebra, mainly functions. By exploring students' learning experiences, this study shows that students have some learning obstacles, including ontogenic obstacles due to students' lack of prerequisite knowledge about the concept of variables, didactical obstacles due to the teacher's teaching implementation focusing solely on the operational rather than the structural conception of functions, and epistemological obstacles due to students' limited knowledge in the concept of variables and functions. Therefore, the identified learning obstacles can be one of the references when developing a lesson design about functions for enhancing students' functional thinking ability [ABSTRACT FROM AUTHOR] more...
- Published
- 2023
29. Desenvolvimento do pensamento algébrico e estudo de padrões e regularidades com crianças: perscrutando possibilidades para educação infantil e anos iniciais do ensino fundamental
- Author
-
Sara Miranda de Lacerda and Natália Gil
- Subjects
Base Nacional Comum Curricular ,Early Algebra ,pensamento algébrico ,raciocínio abstrato ,Theory and practice of education ,LB5-3640 - Abstract
Resumo: O objetivo deste artigo é analisar as possibilidades pedagógicas para um ensino que favoreça o desenvolvimento do pensamento algébrico desde os primeiros anos, assumindo a Base Nacional Comum Curricular (BNCC) como ponto de partida e pautado em literatura recente acerca do tema. Trata-se de um ensaio em que são apresentados alguns aportes teóricos os quais evidenciam que a melhora das capacidades de raciocínio e comunicação de ideias progressivamente mais abstratas está relacionada com desafios e explorações a que os alunos são expostos desde bem pequenos em atividades que valorizam a criatividade e a resolução de problemas. Destacamos, ainda, que vários pesquisadores têm indicado a importância do estudo de padrões e regularidades em Early Algebra, considerando o desenvolvimento do pensamento algébrico como etapa fundamental para o aprendizado da álgebra. O texto apresenta algumas ideias de trabalho pedagógico com padrões e regularidades para a educação infantil e os anos iniciais do ensino fundamental com base nas indicações da BNCC e aponta para a importância de destacar esse tema na formação de professores. more...
- Published
- 2022
- Full Text
- View/download PDF
30. The Equals Sign – the Problem of Early Algebra Learning and How to Solve It
- Author
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Nenad S. Milinković and Sanja M. Maričić
- Subjects
contextual approach ,equals sign ,early algebra ,realistic mathematics education ,student achievement ,Education - Abstract
The equals sign is one of the most important concepts and symbols in mathematics, the understanding of which is crucial for learning and understanding all mathematical content, especially algebra. The paper will draw attention to the problems related to the formation and understanding of this concept, and present a methodological approach to learning based on the context modeling of real-life situations (modeling length, balance, etc.), with the aim of overcoming this misconception. In the empirical section of the paper, we examined the effects of the methodological framework of learning algebra on the understanding of the equals sign through an experiment with parallel groups, and on a sample of the fourth-grade students of primary school (N = 257). The obtained results show that the methodological approach based on the context modeling of real-life situations improves student understanding of the equals sign, as a symbol of mathematical equivalence and their ability to solve problems containing this sign. more...
- Published
- 2022
- Full Text
- View/download PDF
31. LEARNING REPORTED BY THREE TEACHERS AND THE TEACHING OF ALGEBRA IN THE FIRST GRADES.
- Author
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Baldan da Silva, Daniela Inês, Ribeiro, Alessandro Jacques, and Aguiar, Marcia
- Subjects
- *
ALGEBRA education in elementary schools , *TEACHER development , *FIRST grade (Education) , *PROFESSIONAL employee training , *IN-service training of teachers , *TEACHER education , *PROFESSIONAL education , *GROUNDED theory , *TEACHER educators , *TEACHING methods - Abstract
In this paper, our aim is to understand how teachers perceive their professional learning in an in-service teacher education process, as well as the way they related this learning to their teaching practices regarding algebraic thinking in the first grades of elementary school. It is a qualitative-interpretative study, whose data were analyzed considering the Grounded Theory. The results indicate signs of professional learning identified in teaching practice were a glimpse of possibilities to approach algebraic thinking in first grades; changing the planning for students to select and develop mathematical tasks; adopting new methodological strategies with elements of the exploratory teaching approach. [ABSTRACT FROM AUTHOR] more...
- Published
- 2023
- Full Text
- View/download PDF
32. From Symbolic Manipulations to Stepwise Instructions: A Curricular Comparison of Swedish School Algebra Content over the Past 40 Years.
- Author
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Bråting, Kajsa
- Subjects
- *
CURRICULUM , *MATHEMATICS education , *CONTENT analysis , *HIGHER education , *ADULTS - Abstract
Although there have been a huge number of attempts to improve school algebra teaching, several countries are still struggling to improve students algebraic skills. In this study, we focus on the specific case of Sweden where students for several decades have had major problems mastering algebra. In order to get a better understanding of the Swedish situation, we consider what constitutes Swedish school algebra by investigating the development of algebraic content in the Swedish mathematics curriculum documents over the past 40 years. The results reveal that the connection between arithmetic and algebra, the so-called generalized arithmetic, is almost absent in all three curricula although researchers argue that generalized arithmetic is one of the most relevant topics within early algebra. Instead, Sweden has chosen a unique approach as programming, with a specific focus on stepwise instructions and algorithms, recently has been implemented within the core content of algebra. [ABSTRACT FROM AUTHOR] more...
- Published
- 2023
- Full Text
- View/download PDF
33. Τwo different types of technologically enhanced intervention modules to support early algebraic thinking.
- Author
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Chimoni, Maria, Pitta-Pantazi, Demetra, and Christou, Constantinos
- Subjects
ALGEBRA ,PRE-tests & post-tests ,ARITHMETIC ,MODERN language education ,EDUCATIONAL technology - Abstract
This study investigated the role of online applets in early algebra lessons. The effect of two different types of intervention modules on developing students' early algebraic thinking abilities was compared. The first intervention module involved the use of open applets and real-life contexts (open-real). The second intervention module involved the use of closed applets and pure mathematics contexts (closed/pure). "Open" applets are considered to promote more explorative ways of working with mathematical ideas, whereas "closed" applets are considered to guide students' ways of working through more sequential, step-by-step approaches. Real-life contexts present everyday applications of mathematics, where pure mathematics contexts focus on the mathematical concepts and procedures, with no reference to the way they could be associated with real-life situations. Nevertheless, both intervention modules followed an inquiry-based approach. The total number of the participants were 96 young students of Grade 5 with an average age of 10,5 years old. These students were tested through a pre- and a post-test on early algebraic thinking. The test involved three categories of early algebra tasks: generalized arithmetic, functional thinking, and modeling languages. Data from the pre- and post-test comparison showed that students who participated in the "open/real" module had a statistically significant higher improvement in functional thinking compared to students who participated in the "closed/pure" module. There were no statistically significant differences between the improvement of the two groups of students in generalized arithmetic and modeling languages. These findings offer pedagogical implications in respect to the design of early algebra lessons that take advantage of the affordances of available educational technology. [ABSTRACT FROM AUTHOR] more...
- Published
- 2023
- Full Text
- View/download PDF
34. Performance and Strategies Used by Elementary School Fifth Graders When Solving Problems Involving Functional Reasoning
- Author
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Teixeira, Antonio César Nascimento, Magina, Sandra Maria Pinto, Merlini, Vera Lucia, Spinillo, Alina Galvão, editor, Lautert, Síntria Labres, editor, and Borba, Rute Elizabete de Souza Rosa, editor
- Published
- 2021
- Full Text
- View/download PDF
35. A Kindergarten Student’s Use and Understanding of Tables While Working with Function Problems
- Author
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Brizuela, Bárbara M., Blanton, Maria, Kim, Yangsook, Spinillo, Alina Galvão, editor, Lautert, Síntria Labres, editor, and Borba, Rute Elizabete de Souza Rosa, editor
- Published
- 2021
- Full Text
- View/download PDF
36. Exploring quantitative relationship through area conservation activity
- Author
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Puspita Sari and Swee Fong Ng
- Subjects
area conservation ,area measurement ,early algebra ,quantitative relationship ,Mathematics ,QA1-939 - Abstract
Algebra as a study of quantitative relationship is one of four conceptions of school algebra which serves as a foundation for the concept of function. However, there is still a lack of attention to this particular relationship, especially in early algebraic reasoning. This study aims to investigate how the aspect of quantitative relationship in early algebra can be explored through area conservation activities. Understanding area conservation is said to be fundamental in developing the concept of area measurement. In this study, a ten-year old pupil was observed during her involvement while comparing area of two polygons that can be decomposed into equivalent triangles. Data for this study include the pupil’s written artefacts, and video recordings of the activities and interviews. Findings from this study show that the area conservation activity has the potential to build the notion of quantitative relationships in early algebra. The quantitative relationship between the unit of measurement and the result of measurement of a shape can also be explored, that is, the smaller the unit of measurement, the larger the result of measurement. Hence, this study can provide a groundwork for further studies in the relation between quantitative relationship in algebra and area conservation in geometry at the elementary school level. more...
- Published
- 2022
- Full Text
- View/download PDF
37. Connecting characterizations of equivalence of expressions: design research in Grade 5 by bridging graphical and symbolic representations.
- Author
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Tondorf, Alexandra and Prediger, Susanne
- Subjects
- *
DESIGN research , *MATHEMATICAL equivalence , *MATHEMATICAL variables , *MATHEMATICS students , *MATHEMATICS education , *MATHEMATICS teachers - Abstract
One typical challenge in algebra education is that many students justify the equivalence of expressions only by referring to transformation rules that they perceive as arbitrary without being able to justify these rules. A good algebraic understanding involves connecting the transformation rules to other characterizations of equivalence of expressions (e.g., description equivalence that both expressions describe the same situation or figure). In order to overcome this disconnection even before variables are introduced, a design research study was conducted in Grade 5 to design and investigate an early algebra learning environment to establish stronger connections between different mental models and representations of equivalence of expressions. The qualitative analysis of design experiments with 14 fifth graders revealed deep insights into complexities of connecting representations. It confirmed that many students first relate the representations in ways that are too superficial without establishing deep connections. Analyzing successful students' processes helped to identify an additional characterization that can support students in bridging the connection between other characterizations, which we call restructuring equivalence. By including learning opportunities for restructuring equivalence, students can be supported to compare expressions in graphical and symbolic representation simultaneously and dynamically. The design research study disentangles the complex requirements for realizing the design principle of connecting multiple representations, which should be of relevance beyond the specific concept of equivalence and applicable to other mathematical topics. [ABSTRACT FROM AUTHOR] more...
- Published
- 2022
- Full Text
- View/download PDF
38. Starting points: understanding children's pre-instructional intuitions about function tables.
- Author
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Xolocotzin, Ulises, Medrano-Moya, Ana M., and Rojano, Teresa
- Subjects
INTUITION ,GENERALIZATION ,GRADING of students ,ALGEBRA ,ARITHMETIC - Abstract
Functional thinking is an established route into algebra. However, the learning mechanisms that support the transition from arithmetic to functional thinking remain unclear. In the current study we explored children's pre-instructional intuitive reactions to functional thinking content, relying on a conceptual change perspective and using mixed methods. The sample included 20 grade 3 students and 24 grade 5 students. First, we assessed children's arithmetic skills and intuitive responses to generalisation tasks involving variation tables. The quantitative analysis showed that students' arithmetic skills correlated with functional aspects such as the following: identifying, and expressing function rules with words but not with the symbolic expression of function rules. The qualitative analysis revealed that students constructed framework theories that generated different intuitive conceptions of the algebraic ideas involved in noticing and expressing generalisation. Students' conflicts were concentrated in areas that determine key differences between arithmetic and algebra, such as generalisation, indeterminate quantities, and variable notation. We discuss how these results contribute to explaining the construction of algebra concepts. [ABSTRACT FROM AUTHOR] more...
- Published
- 2022
- Full Text
- View/download PDF
39. Developing algebraic activity through conjecturing about relationships.
- Author
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Coles, Alf and Ahn, Aehee
- Subjects
LOGICAL prediction ,WORKING class - Abstract
This manuscript contributes to research on how algebraic thinking about operations and properties can develop, and relevant forms of curricular activity. The key question asked is: how can students' early algebraic activity be fostered through focusing on relations involving operations and properties? We adopt Radford's three aspects of algebraic thinking (indeterminacy, denotation and analyticity), which focus on considering, symbolising and operating on unknown objects. We investigate the work of a class of 11–12 year old students in re-analysing data collected over 20 years ago as part of a project exploring algebraic activity. Our findings point to the proposal that a focus on conjecturing about relationships can be a powerful route into early algebra. On the basis of our analysis we propose an extension to Radford's third aspect of algebraic thinking (analyticity) to include structuring a mathematical space, e.g., through a set of conjectures. [ABSTRACT FROM AUTHOR] more...
- Published
- 2022
- Full Text
- View/download PDF
40. The role of variables in relational thinking: an interview study with kindergarten and primary school children.
- Author
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Lenz, Denise
- Subjects
KINDERGARTEN children ,SCHOOL children ,PRIMARY schools ,NUMBER concept - Abstract
Relational thinking and dealing with variables are two essential aspects of algebraic thinking. Relational thinking means viewing mathematical expressions and equations as a whole rather than as individual computing processes. It is characterized by using relationships between mathematical objects, and refers to the relations of equality and inequality. In this study, to examine the relational thinking of kindergarten and primary school children, this perspective was applied using non-symbolic representations in the form of boxes and marbles. Using multiple variables is a very powerful but also difficult tool of algebra. The study had the aim of examining how kindergarten children and primary school children establish relationships between several variables which are represented with real materials. The interview study was conducted with children aged 5–10 years. Marbles and different colored boxes represented equations with unknowns and quantities depending on each other. Initially, two approaches could be differentiated, namely, number-oriented and structure-oriented approaches. It could be shown that certain conceptualizations of variables were related to children's ability to show relational thinking. Kindergarteners are stimulated to think relationally by unknown quantities which can be determined. This process was observed in primary school children dealing with quantities that depended on each other. In addition, the conceptualization of the variables represented as boxes was examined. The concepts of general number and variable as changing quantity were categorized. Further conceptualizations resulted from the interview data, namely, categories of the undeterminable, the specific number, and the quasi-general. [ABSTRACT FROM AUTHOR] more...
- Published
- 2022
- Full Text
- View/download PDF
41. Fifth graders' learning to solve equations: the impact of early arithmetic strategies.
- Author
-
Xie, Shengying and Cai, Jinfa
- Subjects
EQUATIONS ,ACADEMIC achievement ,PRE-tests & post-tests ,ARITHMETIC - Abstract
In this study we aimed to inquire into the impact of the use of early arithmetic strategies by a group of fifth-grade students, on their solving of equations involving two representations of unknowns. Pre- and post-tests consisting of equation-solving items involving two representations of unknowns (number sentences containing empty 'brackets', as in the example 5 + (∙) = 10, or equations containing x, as in 5 + x = 10), were administered to 126 fifth-grade students in a regular class setting. We found a notable difference between students' success rates on these two types of equations and their strategy use. Most students used the inversing strategy (arithmetic operations) after formal instruction on equation solving. Several students even used both the inversing and formal strategies (performing the same operation on both sides) for the same equation. When the unknown x appeared as the subtrahend or the divisor, the success rate dropped dramatically, and students tried to use the formal solving method of performing the same operation on both sides to solve such equations. The findings of this study not only suggest how teachers can be sensitive to students' different interpretations of unknowns, but also highlight the importance of using students' prior sense making to teach equation solving and of helping students gain an in-depth understanding of equation solving and representations of unknowns. [ABSTRACT FROM AUTHOR] more...
- Published
- 2022
- Full Text
- View/download PDF
42. Generalisation in students with autism spectrum disorder: an exploratory study of strategies.
- Author
-
Goñi-Cervera, J., Cañadas, M. C., and Polo-Blanco, I.
- Subjects
CHILDREN with autism spectrum disorders ,AUTISM spectrum disorders ,GENERALIZATION ,SCHOOL children ,EDUCATION students ,PRIMARY education - Abstract
Generalisation is a skill that enables learners to acquire knowledge in general, and mathematical knowledge in particular. It is a core aspect of algebraic thinking and, in particular, of functional thinking, as a type of algebraic thinking. Introducing primary school children to functional thinking fosters their ability to generalise, explain and reason with mathematical relationships. It also helps them overcome difficulties in understanding functions when they are exposed to the idea more formally in secondary education. Although more and more special education students are enrolled in mainstream schools, little is known about algebraic thinking in that community, especially in the case of students with autism spectrum disorder (ASD). Students with ASD often exhibit deficits that interfere directly with mathematical learning. The study discussed here, which was conducted in the context of algebraic reasoning, was aimed primarily at identifying and describing the strategies and representations observed in 26 ASD primary education students when performing a task that involved a linear function, and describing the generalisations they performed. The 26 participants were enrolled in 19 mainstream Spanish schools. The tools used, a questionnaire and semi-structured interview, were designed to explore their ability to generalise in a problem involving the function f(x) = 2x + 2. The strategies identified included: (a) bald answering; (b) modelling with manipulatives; (c) drawing; (d) counting and (e) operating. The strategy most frequently observed was operating, represented verbally or symbolically, followed by drawing. Only three students generalised but did not reach the highest level of functional thinking, namely, 'functions as objects'. The results are compared with findings for mainstream students of similar ages. Conjectures around the possible relationships between some findings and the type of thinking characteristic of autism spectrum disorder are put forward. The results carry implications for research with and teaching of students with ASD. [ABSTRACT FROM AUTHOR] more...
- Published
- 2022
- Full Text
- View/download PDF
43. Fourth graders' expression of the general case.
- Author
-
Ayala-Altamirano, Cristina, Molina, Marta, and Ambrose, Rebecca
- Subjects
PROBLEM solving ,NATURAL languages ,GENERALIZATION - Abstract
This study forms part of a classroom teaching experiment on the development of a group of 25 9–10 years old students' algebraic thinking. More specifically, it explored their reasoning while solving word problems built around functional relationships to determine how they generalized through questions posed using natural language, drawn figures or the keyword 'many'. Their written and oral answers to those questions were analyzed qualitatively to determine which approach most effectively supported the expression of their generalization. The results reveal the benefits of posing questions about the general case in different ways while teaching students to use conventional algebraic representations. According to these findings, representing indeterminate quantities with the keyword 'many' induces generalization more successfully than representing them with letters. The use of letters prompts students to seek meaning for the letters, either conventionally, as an unknown or variable quantity, or otherwise, as a label or specific values assigned according to their own criteria. Identifying the most effective procedures may help teachers and curriculum designers formulate mathematical tasks that encourage students to express the generality they perceive in particular cases. Determining the communication demands of each approach is likewise highly useful. [ABSTRACT FROM AUTHOR] more...
- Published
- 2022
- Full Text
- View/download PDF
44. The use of cultural contexts for patterning tasks: supporting young diverse students to identify structures and generalise.
- Author
-
Hunter, Jodie and Miller, Jodie
- Subjects
CHILD development ,STUDENT engagement ,ACTIVE learning ,STUDENT publications ,TASKS - Abstract
A key aspect of young children's development of algebraic reasoning is the process of visualising and identifying structures to both abstract and generalise. There has been a growing body of research focused on how students form generalisations, this article adds to the existing body of research by examining how young culturally diverse students identify mathematical structures in contextual growing patterns and the teaching and learning actions that assist them to generalise. Data were collected from one classroom of 29 Year Two (6 years old) students in a low socio-economic school in New Zealand. Results from the analysis of lessons related to two tasks showed that the contextual tasks led students to notice different mathematical structures. Specific pedagogical actions were used to facilitate students' engagement with the growing patterns. These included positioning students to engage with different representations (pictorial and numerical, tabular, and natural language) to represent thinking, the use of classroom discussions, noticing and responding to student thinking, and pressing students to find far terms. The findings highlight that both the contextual patterning tasks and teacher actions supported the young students to develop a range of sophisticated generalisations related to the underlying mathematical structure and functional relationships of the growing patterns. [ABSTRACT FROM AUTHOR] more...
- Published
- 2022
- Full Text
- View/download PDF
45. Variables in early algebra: exploring didactic potentials in programming activities.
- Author
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Kilhamn, C., Bråting, K., Helenius, O., and Mason, J.
- Subjects
ALGEBRA ,MATHEMATICAL programming ,MATHEMATICS teachers ,PRAXIS (Process) ,CHILD development ,MATHEMATICS - Abstract
In this paper we consider implications of the current world-wide inclusion of computational thinking in relation to children's development of algebraic thinking. Little is known about how newly developed visual programming environments such as Scratch could enhance early algebra learning. The study is based on examples of programming activities used by mathematics teachers in Sweden, teaching students aged 10–12 years during the first two years of implementing programming in the mathematics curriculum. Informed by Chevallard's praxeology in terms of praxis and logos, we describe, unpack, discuss and expand these activities. Core issues related to algebra found in the three activities are as follows: making implicit variables explicit; using a counter variable; and identifying parameters as a specific type of variable. Our findings show that, in addition to already identified uses of variables in early algebra, programming activities in the early years bring in new aspects and new ways of treating variables that could, potentially, enhance students' understanding of variables and generalization, provided that programming praxis is embedded in an appropriate algebra logos. [ABSTRACT FROM AUTHOR] more...
- Published
- 2022
- Full Text
- View/download PDF
46. The centrality of student-generated representation in investigating generalizations about the operations.
- Author
-
Schifter, Deborah and Russell, Susan Jo
- Subjects
GENERALIZATION ,CENTRALITY ,ARITHMETIC ,ALGEBRA - Abstract
This article addresses the nature of student-generated representations that support students' early algebraic reasoning in the realm of generalized arithmetic. We analyzed representations created by students for the following qualities: representations that distinguish the behavior of one operation from another, that support an explanation of a specific case of a generalization, and that support justification of a generalization. One key finding is that representations in the form of pictures, diagrams, arrangements of manipulatives, or story contexts that embody the meaning of the operation(s) allow students to distinguish between operations. Such representations can be used by young students to illustrate relationships conveyed in specific instances of a general claim. Further, extending these representations to class of numbers is a mechanism for proving a general claim. [ABSTRACT FROM AUTHOR] more...
- Published
- 2022
- Full Text
- View/download PDF
47. INCORPORACIÓN DEL ÁLGEBRA TEMPRANA EN EDUCACIÓN INFANTIL: UN ANÁLISIS DESDE LOS LIBROS DE TEXTO.
- Author
-
Pincheira, Nataly, Acosta, Yeni, and Alsina, Ángel
- Subjects
- *
EARLY childhood education , *TEXTBOOKS , *CONTENT analysis , *MATHEMATICS education , *ALGEBRA - Abstract
This study analyses the mathematical tasks on early algebra in a collection of eight widely distributed Chilean textbooks for Early Childhood Education (4 to 6 years old). The research followed a qualitative methodology, of an exploratory-descriptive nature, using the technique of content analysis. The results show a presence of algebraic tasks in all the textbooks analysed, with a predominance of tasks linked to establishing relations based on the recognition of attributes, followed by tasks that require seriation based on patterns of repetition, and a scarce presence of tasks that involve the description of changes. [ABSTRACT FROM AUTHOR] more...
- Published
- 2022
- Full Text
- View/download PDF
48. Evaluación del conocimiento para enseñar álgebra temprana durante la formación inicial del profesorado de Educación Infantil.
- Author
-
Pincheira, Nataly and Alsina, Ángel
- Subjects
EARLY childhood teachers ,EARLY childhood education ,STUDENT teachers ,ELECTRONIC textbooks ,MATHEMATICS education ,JUDGMENT (Psychology) ,CURRICULUM - Abstract
Copyright of Revista de Investigación en Educación is the property of Universidad de Vigo, Facultad de Ciencias de la Educacion y del Deporte and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) more...
- Published
- 2022
- Full Text
- View/download PDF
49. An Early Algebra Teaching Experiment Aiming at the Improvement of Fourth-Grade Students' Generalization and Justification Strategies.
- Author
-
YÜCE, Seda and ŞEN, Ceylan
- Abstract
This study aimed to improve fourth-grade students' generalization and justification strategies, and an early algebra teaching experiment was applied for this purpose. In the study, data were obtained through focus group interviews, student worksheets, and video and audio recordings during the research. The data obtained were analyzed by descriptive analysis. In the study, students carried out activities of continuing the pattern, expanding it, completing the missing element, and identifying the pattern rule. It is observed that students used written, verbal, mathematical, and visual representations in their pattern activities. As a result of the study, it was found that students reached generalizations using counting, modeling, iterative, whole-object, contextual, and linear strategies in the teaching experiment. Moreover, it is seen that students used explanation, figural, numerical, and algebraic strategies to verify their generalizations. In line with these results, it is concluded that the early algebra teaching experiment performed with primary school fourth-grade students is effective in students' generalization and justification strategies. [ABSTRACT FROM AUTHOR] more...
- Published
- 2022
- Full Text
- View/download PDF
50. Searching for buried treasure: uncovering discovery in discovery-based learning
- Author
-
Chase, Kiera and Abrahamson, Dor
- Subjects
Basic Behavioral and Social Science ,Behavioral and Social Science ,Clinical Research ,Design-based research ,Discovery learning ,Early algebra ,Technology ,Curriculum and Pedagogy ,Specialist Studies in Education ,Psychology ,Education - Abstract
Forty 4th and 9th grade students participated individually in tutorial interviews centered on a problem-solving activity designed for learning basic algebra mechanics through diagrammatic modeling of an engaging narrative about a buccaneering giant burying and unearthing her treasure on a desert island. Participants were randomly assigned to experimental (Discovery) and control (No-Discovery) conditions. Mixed-method analyses revealed greater learning gains for Discovery participants. Elaborating on a heuristic activity architecture for technology-based guided-discovery learning (Chase and Abrahamson 2015), we reveal a network of interrelated inferential constraints that learners iteratively calibrate as they each refine and reflect on their evolving models. We track the emergence of these constraints by analyzing annotated transcriptions of two case-study student sessions and argue for their constituting role in conceptual development. more...
- Published
- 2018
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