Proposing and testing a model to explain traits of algebra preparedness

Venenciano, Linda; Heck, Ronald H.

doi:10.1007/s10649-015-9672-5

Cited by 11 publications

(7 citation statements)

References 15 publications

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“…Algebraic thinking is often focused on functional thinking, generalisation of patterns, and work on variables (Blanton et al, 2019;Gutierrez et al, 2016). Developing algebraic thinking and number sense of young students, 5 to 12 years old, may also be connected to the analysis of arithmetical structures as a foundation of algebra (Carraher & Schlieman, 2007;Kaput, 2008;Kieran et al, 2016;Radford, 2014;Venenciano & Heck, 2016;Warren, Trigueros, & Ursini, 2016). A complementary explanation of algebraic thinking states that the core of algebraic thinking is the analysis of relationships (Bourbaki, 1974;Davydov, 1982;Krutetskii, 1976).…”

Section: Algebraic Thinkingmentioning

confidence: 99%

Learning actions indicating algebraic thinking in multilingual classrooms

Eriksson

2020

Educ Stud Math

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This article discusses algebraic thinking regarding positive integers and rational numbers when students, 6 to 9 years old in multilingual classrooms, are engaged in an algebraic learning activity proposed by the El’konin and Davydov curriculum. The main results of this study indicate that young, newly arrived students, through tool-mediated joint reflective actions as suggested in the ED curriculum, succeeded in analysing arithmetical structures of positive integers and rational numbers. When the students participated in this type of learning activity, they were able to reflect on the general structures of numbers established as additive relationships (addition and subtraction) as well as multiplicative relationships (multiplication and division) and mixtures thereof, thus a core foundation of algebraic thinking. The students then used algebraic symbols, line segments, verbal, written, and gesture language to elaborate and construct models related to these relationships. This is in spite of the fact that most of the students were second language learners. Elaborated in common experiences staged in the lessons, the learning models appeared to bridge the lack of common verbal language as the models visualized aspects of the relationships among numbers in a public manner on the whiteboard. These learning actions created rich opportunities for bridging tensions in relation to language demands in the multilingual classroom.

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Section: Algebraic Thinkingmentioning

confidence: 99%

Learning actions indicating algebraic thinking in multilingual classrooms

Eriksson

2020

Educ Stud Math

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“…Focus on structural relation Davydov's method of comparing concrete quantities and then representing the comparison algebraically in a range of structural representations of the comparative relation differs from the approaches suggested by other research traditions for additive problem-solving. This emphasis on structural relation between quantities rather than enumerating quantities has been taken up in several Davydov-inspired studies internationally (Polotskaia, 2017;Venenciano & Heck, 2016). The attention to multiple ways of representing single additive relations contrasts with the problem-solving approaches promoted, for example, in Carpenter et al's (1999) work on additive situations.…”

Section: Davydov's Approach To Early Number Teachingmentioning

confidence: 99%

“…One way of demarcating the international Davydov-related incorporations of early number teaching is to divide those structure-oriented approaches that have taken on the non-numerical introduction to quantitative comparison, from approaches that have adopted the attention to structure but broken with the algebraic representation to use numerical representations of quantity instead. Jean Schmittau's implementation of Davydov's curriculum in the USA (see Morris, 2000;Schmittau & Morris, 2004) and the work of the Measure Up team in Hawaii (Dougherty & Slovin, 2004;Venenciano & Heck, 2016) both fall into the first fidelity of implementation category. In contrast, in Polotskaia's (2017) work in Canada and Mellone and Tortora's (2017) work with older children in Italy, numbers are used in structural comparison activities.…”

Section: Davydov's Approach To Early Number Teachingmentioning

confidence: 99%

Shape-shifting Davydov’s ideas for early number learning in South Africa

2020

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In this paper, we share details of a South African early grades’ number intervention informed by aspects of Davydov’s writing on early number teaching and learning. A key part of Davydov’s approach to early number teaching involves starting with attention to relationships between quantities rather than with counting. The Structuring Number Starters (SNS) intervention focused—over a nine-year period—on supporting early grades’ students to move beyond the calculating-by-counting approaches that are prevalent in South Africa. In attending to this focus, the intervention shifted increasingly towards an emphasis on relationships between quantities, though not in the same format or task sequence as advocated by Davydov. The contextual and cultural features that led to our adaptations—or shape-shifting—are highlighted in this paper. We interrogate key aspects of Davydov’s approaches to early number teaching in relation to key features typical of South African classroom mathematics teaching in order to understand the evolution of the SNS initiative. Quasi-longitudinal interview-based assessment data available from a cross-attainment sample of students in 2011, 2014 and 2018 indicate shifts over time from calculating-by-counting to calculating-by-structuring. These outcomes point to successes with moves into increasingly structured ways of working with early number, but suggest also that these successes may be contingent on some fluency with forward and backward number word sequences. The outcomes suggest that it is feasible to explore interventions directing attention to early number structure from the outset in larger scale studies.

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“…The relation 3 Other researchers have developed and investigated instructional approaches based on the measurement of continuous quantities. See, for example, the work of Brousseau, Brousseau, and Warfield (2004); Carraher (1996); Davydov and Tsvetkovich (1991); Dougherty and Venenciano (2007); Powell (2019); Venenciano and Heck (2016); and Venenciano, Slovin, and Zenigami (2015).…”

Section: Measuring Perspectivementioning

confidence: 99%

How does a fraction get its name?

Powell¹

2019

Rev. Bra. Edu. Ciê. Edu. Mat.

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Philosophical and cultural perspectives shape how a fraction is named and defined. In turn, these perspectives have consequences for learners' conceptualization of fractions. We examine historical foundations of two perspectives of what are fractions—partitioning and measuring—and how these views influence fraction knowledge. For the dominant perspective, partitioning, we indicate how its approach to what is a fraction that discretizes objects and its well-meaning visual correlates cause learners a host of perceptual difficulties. Based on the human cultural and social practice of measuring continuous quantities, we then offer an alternative understanding of what is a fraction and illustrate the promise of this view for fraction knowledge. We introduce pedagogical tools, Cuisenaire rods, and illustrate how they can be used to implement a measuring perspective to comprehending properties and a definition of fractions. We end by sketching how to initiate a measuring perspective in a mathematics classroom.Keywors: Fractions; Gattegno; Measuring; Partitioning; Unit fractions. Como uma fração recebe seu nome?Resumo: Perspectivas filosóficas e culturais moldam como uma fração é nomeada e definida. Por sua vez, essas perspectivas têm consequências para a conceitualização de frações dos estudantes. Examinamos os fundamentos históricos de duas perspectivas do que são frações—particionamento e medição—e como essas visões influenciam o conhecimento das frações. Para a perspectiva dominante, partição, indicamos como sua abordagem ao que é uma fração que discretiza objetos e seu correlato visual bem-intencionado causa aos alunos uma série de dificuldades perceptivas. Com base na prática cultural e social humana de medir quantidades contínuas, oferecemos um entendimento alternativo do que é uma fração e ilustramos a promessa dessa visão para o conhecimento da fração. Introduzimos ferramentas pedagógicas, varas Cuisenaire e ilustramos como elas podem ser usadas para implementar uma perspectiva de medição para compreender propriedades e uma definição de frações. Terminamos esboçando como iniciar uma perspectiva de medição em uma sala de aula de matemática.Palavras-chave: Frações; Gattegno; Medição; Partição; Frações unitárias.

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Proposing and testing a model to explain traits of algebra preparedness

Cited by 11 publications

References 15 publications

Learning actions indicating algebraic thinking in multilingual classrooms

Learning actions indicating algebraic thinking in multilingual classrooms

Shape-shifting Davydov’s ideas for early number learning in South Africa

How does a fraction get its name?

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