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A study investigated the impact of a sustained, comprehensive early algebra intervention in third grade. The authors share and discuss students' responses to a written pre- and post-assessment that addressed their understanding of several big ideas in the area of early algebra, including mathematical equivalence and equations, generalizing arithmetic, and functional thinking.

This study aims to investigate pre-service mathematics teachers’ (PMTs) reflections on the 2018-2020 High School Entrance (HSE) exam questions and the characteristics of the algebra tasks developed by the PMTs after participating in an intervention that involves examining and categorizing algebra questions. It also examines the degree to which PMT-generated tasks are cognitively demanding and consistent with the characteristics of HSE exam questions. A case study was employed in this study as a qualitative research design. The study was conducted in the 2020-2021 Spring semester as part of the Methods of Teaching Mathematics in Middle Schools II course offered in the 3rd year of a 4-year teacher education program in a public university. A total of 29 PMTs enrolled in the course, and they were asked to work in groups. The study intervention took place during the course, and one of these groups was focused on during the intervention. The data were collected through two algebra tasks that the group produced at the end of the intervention, four PMTs’ individual responses to a written form, and their semi-structured individual interviews about reflections on HSE exam questions. All the collected data were qualitatively analyzed using content analysis. Both existing categories in the literature and new emergent categories were used in this process. Four categories emerged from PMTs’ reflections on HSE exam algebra questions: algebra objectives, use of algebra, use of context, and cognitive demand after participating in the intervention. The findings about the characteristics of the tasks developed by PMTs indicated that they were capable of developing cognitively demanding algebra tasks, which were mostly consistent with PMTs’ reflections on the characteristics of HSE exam questions. The implications of the findings for PMTs’ reflections on HSE exam questions and the tasks they developed were also discussed. Bu çalışma ortaokul matematik öğretmen adaylarının (MÖA) 2018-2020 Liselere Giriş Sınav (LGS) soruları hakkındaki yansıtıcı düşüncelerini ve cebir sorularının incelenmesi ve kategorize edilmesiyle ilgili bir müdahaleye katıldıktan sonra öğretmen adayları tarafından geliştirilen cebir görevlerinin özelliklerini incelemeyi amaçlamaktadır. Ayrıca, bu çalışma öğretmen adayları tarafından oluşturulan görevlerin bilişsel olarak ne derece zorlayıcı olduğunu ve LGS sorularının özellikleriyle ne derece tutarlı olduğunu da incelemektedir. Bu çalışmada nitel bir araştırma deseni olan durum çalışması kullanılmıştır. Çalışma, 2020-2021 bahar döneminde, bir devlet üniversitesinde 4 yıllık öğretmen eğitimi programının 3. yılında verilen Ortaokullarda Matematik Öğretim Yöntemleri II dersi kapsamında yürütülmüştür. Derse toplam 29 matematik öğretmen adayı kaydolmuştur ve öğretmen adaylarından grup halinde çalışmaları istenmiştir. Müdahale ders sırasında gerçekleşmiş ve müdahale sırasında bu gruplardan birine odaklanılmıştır. Veriler, grubun müdahalenin sonunda ürettiği iki cebir görevi, dört öğretmen adayının yazılı forma verdiği bireysel yanıtlar ve LGS sorularına ilişkin yansıtıcı düşünceleri hakkında yarı yapılandırılmış bireysel görüşmeler yoluyla toplanmıştır. Toplanan tüm veriler içerik analizi kullanılarak nitel olarak analiz edilmiştir. Bu süreçte hem literatürdeki mevcut kategoriler hem de ortaya çıkan yeni kategoriler kullanılmıştır. Öğretmen adaylarının yansıtıcı düşüncelerinden cebir kazanımları, cebir kullanımı, bağlam kullanımı ve bilişsel istem olacak şekilde dört kategori ortaya çıkmıştır. Öğretmen adayları tarafından geliştirilen görevlerin özelliklerine ilişkin bulgular, öğretmen adaylarının bilişsel olarak zorlu cebir görevleri geliştirebildiklerini ve bu görevlerin sınav sorularının özelliklerine ilişkin yansıtıcı düşünceleriyle uyumlu olduklarını göstermiştir. Öğretmen adaylarının sınav soruları hakkındaki yansıtıcı düşünceleri ve geliştirdikleri görevler ile ilgili bulguların sonuçları tartışılmıştır.

A cluster randomized trial design was used to examine the effectiveness of a Grades 3 to 5 early algebra intervention with a diverse student population. Forty-six schools in three school districts participated. Students in treatment schools were taught the intervention by classroom teachers during regular mathematics instruction. Students in control schools received only regular mathematics instruction. Using a three-level longitudinal piecewise hierarchical linear model, the study explored the impact of the intervention in terms of both performance (correctness) and strategy use in students' responses to written algebra assessments. Results show that during Grade 3, treatment students, including those in at-risk settings, improved at a significantly faster rate than control students on both outcome measures and maintained their advantage throughout the intervention.

Mathematics educators have argued for some time that elementary school students are capable of engaging in algebraic thinking and should be provided with rich opportunities to do so. Recent initiatives like the Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010) have taken up this call by reiterating the place of early algebra in children's mathematics education, beginning in kindergarten. Some might argue that early algebra instruction represents a significant shift away from arithmetic-focused content that has typically been taught in the elementary grades. To that extent, it is fair to ask, \"Does early algebra matter?\" That is, will teaching children to think algebraically in the elementary grades have an impact on their algebra understanding in ways that will potentially make them more mathematically successful in middle school and beyond? In this article, the authors share findings from a research project whose goal is to study the impact of a comprehensive early algebra curricular experience on elementary school students' algebraic thinking within a range of domains including generalized arithmetic, equivalence relations, functional thinking, variables, and proportional reasoning. The focus here is on the performance of third-grade students who participated in an early algebra intervention on a written assessment administered before and after instruction. The authors also discuss the strategies these students used to solve particular tasks and provide examples of the classroom activities and instructional strategies that they think supported the growth they saw in students' algebraic thinking. A bibliography is included.

Isler et al talk about the study of functions which has traditionally received the most attention at the secondary level, both in curricula and in standards documents--the Common Core State Standards for Mathematics and Principles and Standards for School Mathematics. The growing acceptance of algebra as a K-grade 12 strand of thinking by math education researchers and in standards documents, along with the view that the study of functions is an important route into learning algebra, raises the importance of developing children's understanding of functions in the elementary grades.

Learning progressions have been demarcated by some for science education, or only concerned with levels of sophistication in student thinking as determined by logical analyses of the discipline. We take the stance that learning progressions can be leveraged in mathematics education as a form of curriculum research that advances a linked understanding of students learning over time through careful articulation of a curricular framework and progression, instructional sequence, assessments, and levels of sophistication in student learning. Under this broadened conceptualization, we advance a methodology for developing and validating learning progressions, and advance several design considerations that can guide research concerned with engendering forms of mathematics learning, and curricular and instructional support for that learning. We advance a two-phase methodology of (a) research and development, and (b) testing and revision. Each phase involves iterative cycles of design and experimentation with the aim of developing a validated learning progression. In particular, we gathered empirical data to revise our hypothesized curricular framework and progression and to measure change in students. thinking over time as a means to validate both the effectiveness of our instructional sequence and of the assessments designed to capture learning. We use the context of early algebra to exemplify our approach to learning progressions in mathematics education with a focus on the concept of mathematical equivalence across Grades 3-5. The domain of work on research on learning over time is evolving; our work contributes a broadened role for learning progressions work in mathematics education research and practice.

We share here results from a quasi-experimental study that examines growth in students' algebraic thinking practices of generalizing and representing generalizations, particularly with variable notation, as a result of an early algebra instructional sequence implemented across grades 3-5. Analyses showed that, while there were no significant differences between experimental and control students on a grade 3 pre-assessment measuring students' capacity for generalizing and representing generalizations, experimental students significantly outperformed control students on post-assessments at each of grades 3-5. Moreover, experimental students were able to more flexibly interpret variable in different roles and were better able to use variable notation in meaningful ways to represent arithmetic properties, expressions and equations, and functional relationships. This study provides important evidence that young children can learn to think algebraically in powerful ways and suggests that the earlier introduction of algebraic concepts and practices is beneficial to students.

Given the emphasis on teaching and learning proof across all grades in the last decade, conceptualizing proof at the elementary level has gained more attention. The purpose of this study is to investigate elementary teachers’ expectations regarding reasoning and proof in school mathematics, their recognition of different kinds of student reasoning, and the expectations they have for their students around reasoning and proof. The data were collected through an online survey and follow-up interviews. The findings of this mixed methods study showed that teachers report that they implement practices that are in line with reform documents to help students learn proof such as asking their students to show why or how and most teachers focused on the role of reasoning and proving as developing understanding. However, not many teachers talked about guiding their students to develop and prove/disprove conjectures. Visual generic example arguments were teachers’ favorite arguments for convincing their students and for utilizing in their instruction; their most commonly stated reasons were the arguments’ visual or tactile features and their accessibility to students in determining the convincingness and utilization.

The study of functions has traditionally received the most attention at the secondary level, both in curricula and in standards documents--for example, the Common Core State Standards for Mathematics (CCSSI 2010) and \"Principles and Standards for School Mathematics\" (National Council of Teachers of Mathematics [NCTM] 2000). However, the growing acceptance of algebra as a K-grade 12 strand of thinking by math education researchers and in standards documents, along with the view that the study of functions is an important route into learning algebra (Carraher and Schliemann 2007), raises the importance of developing children's understanding of functions in the elementary grades. What might it look like to engage students in functional thinking in the elementary grades? Elementary school curricula often include a focus on simple patterning activities (e.g., recursive number sequences, such as 2, 4, 6, 8, …) in which only one variable is observed. However, an exclusive focus on this type of activity might hinder the development of students' reasoning about how two or more quantities vary simultaneously (Blanton and Kaput 2004), a key component of functional thinking. Blanton and her colleagues argue that elementary school students are in fact capable of engaging in this type of thinking (Blanton et al. 2011, p. 47). Furthermore, they point out that focusing on functional thinking provides a context for students to understand the role of variable as varying quantity. This article supports Blanton and her colleagues' (2011) argument by sharing a classroom episode as well as pre-instruction and post-instruction data from a yearlong teaching experiment. The authors discuss some of the crucial elements they believe contributed to students' growing abilities to engage in functional thinking. They also discuss connections made among various representations, another important benefit of having students engage in functional thinking.

Uluslararası alan yazında erken cebirin önemi vurgulanmış ve kapsamlı bir uygulamaya dâhil olan öğrencilerin cebirsel düşünme becerilerini geliştirdikleri birçok çalışmada ele alınmıştır. Bu çalışmanın amacı, bir erken cebir uygulamasının, 3. sınıf öğrencilerin cebirsel düşünme becerilerine etkilerini test etmektir. 3. sınıf uygulama ve kontrol grupları çalışmaya dâhil edilmiştir ve her iki gruba ön ve son test uygulanmıştır. Bulgularda, 3. sınıf uygulama ve kontrol grupları performansları arasında ön testte istatiksel bir fark olmamasına rağmen son testte istatiksel olarak anlamlı bir farklılık bulunmuştur. Öğrencilerin stratejilerine yönelik yapılan analizler, uygulama grubu öğrencilerinin son testte eşitlik, cebirsel ifadeler, denklemler ve eşitsizlikler; genelleştirilmiş aritmetik ve fonksiyonel düşünme üç ana alanında cebirsel düşünmeye dayalı stratejileri kontrol grubuna kıyasla daha çok kullandıklarını ortaya çıkarmıştır.