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Early algebra and algebraic reasoning
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This chapter provides an overview of research about algebraic reasoning among relatively young students (6-12 years), It focuses on mathematics learning and, to a lesser extent, teaching. Issues related to educational policy, epistemology, and curriculum design provide a backdrop for the discussion.
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... As such, Klein (2016) does not support the idea that algebra early should be something else compared to early algebra (e.g., Carraher & Schliemann, 2007). Given how the concept is treated, which to some degree reflects the historical development of the concept function (e.g., González-Polo & Castaneda, 2024;Kjeldsen & Lützen, 2015), the education of functions may create gaps or discontinuity as Klein (2016) puts it. ...
... Hence, there is no chronological pattern between these three definitions; that is, the grey scale becomes more or less intense. Further investigations show that Stephens et al. (2017) rely on the work of Carraher and Schliemann (2007), hence having a different theoretical framing than the other two studies, although using the same materials. Although different ontological framing, the epistemological outcome appears to be consistent. ...
Knowing functions and functional thinking have recently moved from just knowledge for older students to incorporating younger students, and functional thinking has been identified as one of the core competencies for algebra. Although it is significant for mathematical understanding, there is no unified view of functional thinking and how different aspects of the concept are used as a theoretical base. In this paper, we analyse different definitions used in empirical studies. First, we did a systematic research review resulting in 19 empirical studies focusing on functional thinking with an appropriate theoretical underpinning. The definitions were analysed using an AI tool. After that, we analysed the results using intrinsic mathematical properties of how functions can be defined in mathematics to identify core aspects of the definitions. According to the analysis, two definitions capture most of the key aspects of functional thinking, and most empirical studies use these key concepts. These two definitions treat functional thinking as products or products and processes. One definition used in one empirical study stands out by theoretically operationalizing functional thinking as a process. As such, different ontological assumptions are made in the studies; however, in some cases, having the same epistemological outcome. From a methodological point of view, the cosine similarity matrix was a useful tool for an ontological analysis, but a qualitative analysis is still needed to make meaning of it.
... Given calls for M-PSTs to understand mathematics as a coherent field, experiences that support building algebraic connections are essential (AMTE 2017;CBMS, 2012;Cuoco & Rotman, 2013;McCrory et al., 2012). Based on recommendations from policy documents (AMTE, 2017;CBMS, 2012;NCTM 2000NCTM , 2012NGA & CCSSO, 2010) and relevant research (Artzt et al., 2012;Carraher & Schliemann, 2007;Cuoco & Rotman, 2013;Foley, 1997;Gravemeijer, 2002;Kirshner, 2001;Lehrer & Schauble, 2000;McCrory et al., 2012;Moore-Harris, 1998;Usiskin et al., 2002), we describe our conceptual framework which highlights four types of algebraic connections recommended for M-PSTs to encounter in their secondary mathematics teacher education programs (see Figure 1). We align our conceptual framework with prior research focused on vertical and horizontal curricular knowledge (Shulman, 1986;Tyler, 1949) and knowledge for algebra teaching (KAT) (McCrory et al., 2012). ...
... 46). Carraher and Schliemann (2007) presented the referential role of algebra, emphasizing that algebraic knowledge grows out of representing extra-mathematical contexts (Gravemeijer, 2002;Kirshner, 2001;Lehrer & Schauble, 2000). In support of these arguments, NCTM (2000) and AMTE (2017) emphasized that students need opportunities to recognize and apply mathematics in contexts outside of mathematics; in particular, making use of algebra to model and predict real world phenomena. ...
Beginning teachers benefit from preparation emphasizing supporting students with diverse needs and developing deeper understandings of algebraic connections. Our study aims to explore how instructors in secondary preparation programs and mathematics preservice teachers (M-PSTs) conceptualize and enact algebraic connections in required courses. Our dataset comprises instructor interviews and course materials from 48 courses across five universities, and 10 focus group interviews involving 37 M-PSTs. Employing constant comparison method, we analyzed data by creating a coding system for algebraic connections. Our findings highlight varying perspectives of M-PSTs and instructors related to encounters with algebraic connections, emphasis of algebraic connection types by both groups, and types emphasized as main objectives of courses in five secondary mathematics teacher education programs. We describe nine algebraic connection themes that were identified in this investigation. We discuss emergent implications for curriculum development, instructional practices, and foundational algebraic concepts in teacher preparation programs.
... 149) qui considère que : "Algebraic thinking in the early grades involves the development of ways of thinking within activities for which the letter-symbolic could be used as a tool, or alternatively within activities that could be engaged in without using the letter-symbolic at all, for example, analyzing relationships among quantities, noticing structure, studying change, generalizing, problem solving, justifying, proving, and predicting." Dans la même perspective, Carraher and Schliemann [15] caractérisent la pensée algébrique précoce en termes de formes de raisonnements qui expriment des relations entre des nombres ou des quantités, en particulier des relations fonctionnelles. Pour distinguer entre la pensée arithmétique et la pensée algébrique, Radford [13] attribue une place centrale à l'analycité dans la pensée algébrique. ...
Dans ce travail, nous avons cherché, à travers l’exploration des connaissances des enseignants, à étudier leur prédisposition à soutenir le développement de la pensée algébrique (PA) à la fin de l’enseignement primaire tunisien, dans le cadre des activités de résolution de problèmes de partage inéquitable. Nous avons focalisé notre attention sur trois types de connaissances caractérisant la PA élémentaire. (1) l’exploration de la sensibilité des enseignants à la structure déconnectée des problèmes de partage inéquitable, favorisant la mobilisation de raisonnements algébriques. (2) l’étude de la capacité des enseignants à identifier le degré d’analyticité des raisonnements des élèves et à intervenir de manière appropriée pour soutenir le raisonnement algébrique. (3) le troisième type de connaissances est relatif à l’exploration de l’influence du registre de représentation utilisé par les élèves sur la capacité de l’enseignant à évaluer l’analyticité des raisonnements. L’étude empirique a été réalisée via un questionnaire soumis à 53 enseignants de 6ème année de l’enseignement de base tunisien (11 à 12 ans). Les résultats font apparaitre un manque de sensibilité des enseignants à la structure épistémologique des problèmes et aux représentations sémiotiques, révélant des lacunes dans leur formation en algèbre élémentaire et dans les ressources officielles disponibles.
... There is a growing consensus in mathematics education that early algebra should be introduced in primary schools (Blanton et al., 2015a, b;Carraher et al., 2008;Schoenfeld, 1995). Students' experience with early algebra is recognized as a key precursor to the formal study of algebra in middle and high schools (Carpenter et al., 2003;Carraher & Schliemann, 2007;Stephens et al., 2017b). Functional thinking (FT), a main component of algebraic thinking Kaput, 2008), serves as an essential entry point into early algebraic thinking (Blanton & Kaput, 2011;Cañadas et al., 2016;Pinto et al., 2022). ...
Functional thinking has long been recognized as a crucial entry point into algebraic thinking in elementary school. This mixed-method study investigates the learning progression for elementary students’ functional thinking within the context of routine classroom instruction. Drawing on the existing research, a theoretical framework was constructed to assess the functional thinking of 649 students across grades 3 to 6. The framework includes three modes of functional thinking: recursive patterning, covariational thinking, and correspondence relations, each with particular and general layers of algebraic generality. Psychometric analysis was conducted to validate the assessment instrument. The study identifies a five-level learning progression of functional thinking: Pre-Structure, Pre-Functional Thinking, Emergent Functional Thinking, Specific Functional Thinking, and Condensed Functional Thinking. The distinctive characteristics of each level are identified and illustrated. Furthermore, the developmental sequence of different functional thinking modes within this learning progression is analyzed. Finally, the theoretical and practical implications of these findings are examined.
... Algebraic reasoning, an essential aspect of mathematical reasoning, refers to the process of generalizing mathematical ideas through argumentation and formal expression (Blanton & Kaput, 2005). It involves the ability to analyze quantities within contexts and express their relationships through tables, graphs, symbols, and mathematical expressions (Carraher & Schliemann, 2007). Students who develop algebraic reasoning can generalize specific situations into broader mathematical concepts (Kaput, 1999), and it serves as a foundation for learning algebra. ...
Few studies have examined the affective factors influencing students' algebraic reasoning. This study aims to investigate the effect of student engagement and self-regulated learning on algebraic reasoning. The research employed a quantitative correlational design. Using a cluster sampling technique, 202 students from Islamic State junior high schools in Mataram were selected as participants. Data were collected through tests and questionnaires. The instruments used included an algebraic reasoning test, student engagement questionnaires, and self-regulated learning questionnaires. Data analysis involved both descriptive analysis (categorical and statistical) and inferential analysis (prerequisite tests and hypothesis testing). The results of this study indicate that student engagement has a significant influence on algebraic reasoning (t = 2.418, p = 0.017 < 0.05). However, self-regulated learning did not show a significant effect on algebraic reasoning (t = -0.425, p = 0.671 > 0.05). Additionally, student engagement and self-regulated learning, when considered together, significantly influence algebraic reasoning (F = 3.117, p = 0.046 < 0.05). The study also found that student engagement and self-regulated learning account for 3% of the variance in algebraic reasoning (R² = 0.03), with the regression equation Y = 43.491 + 0.277X₁ - 0.06X₂. These findings suggest that teachers should prioritize fostering student engagement and self-regulated learning in the classroom, emphasizing interactive, collaborative, and contextually relevant algebra instruction. Abstrak: Beberapa penelitian telah mengeksplorasi faktor afektif yang memengaruhi penalaran aljabar siswa. Penelitian ini bertujuan untuk mengetahui pengaruh keterlibatan siswa dan pembelajaran yang diatur sendiri terhadap penalaran aljabar. Jenis penelitian yang digunakan adalah kuantitatif korelasional. Dengan teknik sampling klaster, 202 siswa dari Sekolah Menengah Pertama Negeri di Mataram dipilih sebagai sampel penelitian. Data dikumpulkan menggunakan tes dan kuesioner. Instrumen yang digunakan dalam penelitian ini adalah tes penalaran aljabar, kuesioner keterlibatan siswa, dan kuesioner pembelajaran yang diatur sendiri. Analisis data yang digunakan dalam penelitian ini adalah analisis deskriptif (kategori dan statistik deskriptif) dan analisis inferensial (uji prasyarat dan uji hipotesis). Hasil penelitian ini menunjukkan bahwa keterlibatan siswa berpengaruh terhadap penalaran aljabar (t = 2,418, p = 0,017 < 0,05). Namun, pembelajaran yang diatur sendiri tidak berpengaruh terhadap penalaran aljabar (t = -0,425, p = 0,671 > 0,05). Selain itu, keterlibatan siswa dan pembelajaran yang diatur sendiri secara bersamaan berpengaruh terhadap penalaran aljabar (F = 3,117, p = 0,046 < 0,05). Penelitian ini juga menunjukkan bahwa keterlibatan siswa dan pembelajaran yang diatur sendiri memberikan kontribusi sebesar 3% (R² = 0,03) terhadap penalaran aljabar dengan persamaan regresi Y = 43,491 + 0,277X₁ - 0,06X₂. Temuan ini memberikan implikasi bagi guru untuk memprioritaskan keterlibatan siswa dan pembelajaran yang diatur sendiri di dalam kelas, dengan menekankan pembelajaran aljabar yang interaktif, kolaboratif, dan relevan dengan konteks.
... Early algebra has recently been incorporated into the early childhood curriculum (e.g., Australian Curriculum, Assessment and Reporting Authority, 2020; Ministerio de Educación [MINEDUC], 2018; Ministry of Education, Republic of Singapore, 2013;NCTM, 2000). This perspective is intended to introduce the development of algebraic thinking from early ages (Carraher & Schliemann, 2007;Kaput et al., 2008) aiming to ensure a better understanding of mathematics in later educational stages (Cai & Knuth, 2011). The Spanish early childhood curriculum, while including some content associated with early algebra, omits essential topics such as patterns and change (Alsina, 2022). ...
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.
Given the relevance of graphs of functions, we consider their inclusion in primary education from the functional approach to early algebra. The purpose of this article is to shed some light on the students’ production and reading of graphs when they solved generalization problems from a functional thinking approach. We aim to explore how 3rd and 4th graders construct graphs associated to functions and what elements they use; and how they read function associated graphs and whether they connect pairs of values to see beyond the data. After four working sessions about functions, we designed and implemented individual interviews to 12 students. Through a qualitative analysis, we highlight that the students can read data in a graph on two different cognitive levels and also construct it from different elements of the graph initially provided. Regarding data reading, we evidence two levels: (a) literal reading of a given element in the graph, and (b) reading beyond the data. The construction of the graph is described with base on the axes, values and labels on the axes, scale of the axes, and construction techniques. We present examples of students’ work that evidence that graph construction varied depending on whether it was created from a blank sheet or it was necessary to provide help regarding the axes or the scale of the graph. We describe several techniques used by the students in the representation of data that yield non-canonical representations of a graph and that help glimpse how students are interpreting this representation.
Algebra instruction has traditionally been delayed until adolescence because of
mistaken assumptions about the nature of arithmetic and about young students'
capabilities. Arithmetic is algebraic to the extent that it provides opportunities for
making and expressing generalizations. We provide examples of nine-yeur-old
children using algebraic notation to represent a prublern of additive relations. They
not only operate on unknowns; they can understand the unknown to stand for all of
the possible values that an entity can take on. When they do so, they are reasoning
about variables.