“Approximate” multiplicative relationships between quantitative unknowns

Hackenberg, Amy J.; Jones, Robin; Eker, Ayfer; Creager, Mark A.

doi:10.1016/j.jmathb.2017.07.002

Cited by 15 publications

(7 citation statements)

References 33 publications

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“…However, when they do, that does not mean that they have constructed the operations, schemes, and concepts necessary to generate those ideas themselves, and so they may construct different schemes from MC3 students (e.g., Hackenberg & Tillema, 2009). Similarly, MC3 students sometimes cannot see the reason that MC2 students would think the way they do (e.g., Hackenberg et al, 2017), although they can notice and develop an interest in differences (e.g., Hackenberg, 2010a). We used these observations to develop three conjectures about how students might influence each other in discussions in relatively positive ways:…”

Section: Practice 5: Conducting Whole Class Discussion Across Differe...mentioning

confidence: 99%

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Teaching practices for differentiating mathematics instruction for middle school students

Hackenberg

Creager

Eker

2020

Mathematical Thinking and Learning

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Three iterative, 18-episode design experiments were conducted after school with groups of 6-9 middle school students to understand how to differentiate mathematics instruction. Prior research on differentiating instruction (DI) and hypothetical learning trajectories guided the instruction.As the experiments proceeded, this definition of DI emerged: proactively tailoring instruction to students' mathematical thinking while developing a cohesive classroom community. Analysis of 10 episodes across experiments yielded five teaching practices that facilitated DI: using researchbased knowledge of students' mathematical thinking, providing purposeful choices and different pathways, inquiring responsively during group work, attending to small group functioning, and conducting whole class discussions across different thinkers. The latter three practices, at times, impeded DI. This study is a case of using second-order models of students' mathematical thinking to differentiate instruction, and it reveals that inquiring into research-based knowledge and inquiring responsively into students' thinking are at the heart of differentiating mathematics instruction.

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Section: Practice 5: Conducting Whole Class Discussion Across Differe...mentioning

confidence: 99%

“…Similarly, the construction of particular schemes and concepts can dramatically influence how a student interacts with others in a classroom (e.g., Hackenberg, Jones, Eker, & Creager, 2017).…”

Section: Mathematical Thinking and Interactionmentioning

confidence: 99%

Teaching practices for differentiating mathematics instruction for middle school students

Hackenberg

Creager

Eker

2020

Mathematical Thinking and Learning

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“…Hackenberg (2007;2013) and colleagues (2017) have highlighted two operations in fraction schemes as being especially important for algebra: units coordination and disembedding. Specifically, children's ability to coordinate three levels of unitsa unit of composite unitsis important for understanding improper fractions (Hackenberg, 2007) and writing algebraic equations (Hackenberg, 2013;Hackenberg et al 2017;Hackenberg & Lee, 2015;Olive & Caglayan, 2008).…”

Section: Hackenberg (2010) Calls Interiorized Schemes Conceptsmentioning

confidence: 99%

“…In many studies, Hackenberg and others (Byerly, 2019;Hackenberg, 2010;2013;Hackenberg et al, 2017;Hackenberg & Lee, 2015;Kaput & West, 1994;Nabors, 2003;Olive & Çaglayan, 2008;Thompson & Saldhana, 2003) argue that advanced multiplicative conceptsinteriorized fraction schemessupport algebraic reasoning.…”

Section: Near Here]mentioning

confidence: 99%

Causal pathways from fractions to algebra: Integrating psychology and math education perspectives

Viegut¹

2021

Preprint

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Algebra knowledge is an important gatekeeper to educational and economic opportunity. Both math education researchers and psychologists have shown that fractions may be a key to this gate (e.g., Hackenberg, 2013; Siegler et al., 2012). However, psychological and educational research on the fractions-algebra association has been disconnected, with separate frameworks, definitions, and designs. This integrative review synthesizes evidence from both disciplines about how and why fractions knowledge leads to stronger algebra knowledge. I suggest that the strength of causal evidence is limited by idiosyncratic measurement, limited longitudinal research, and imprecise definitions. I also review six plausible fractions-to-algebra mechanisms, which future research should empirically test. Throughout, I argue that more nuanced understanding of the fractions-algebra association will require interdisciplinary teams. Finally, I propose an integrative conceptual model of how fractions knowledge may lead to success in algebra and suggest new directions for collaborative investigation to inform developmental theory and educational practice.

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“…There are several studies in mathematics education literature investigating students' unit coordination stages and how these stages played a role in students' understanding other math topics (e.g., Hackenberg, Aydeniz, Jones, & Borowski, 2017;Hackenberg, Jones, Eker & Creager, 2017;Ulrich & Wilkins, 2017;Zwanch, 2019). Some of these studies pointed out that the difficulty in coordinating two-and three-level units was one of the main reasons for the weak performances in students' mathematics learning (Hackenberg, 2013;Hackenberg & Tillema, 2009;Norton & Boyce, 2013;Steffe & Olive, 2010).…”

Section: Stagementioning

confidence: 99%

Investigation of middle school students’ unit coordination levels in mathematics problems involving multiplicative relations

Acar¹,

Sevinç²

2021

ilkonline

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This study aimed to identify Turkish middle school students' unit coordination levels in mathematics problems involving multiplicative relations and investigate these unit coordination levels in each grade level. This study was designed as a qualitative case study in which a written instrument developed by Norton and his colleagues (2015) was implemented to 139 middle school students. The student responses were coded based on the rubric developed by Norton and his colleagues (2015), and the unit coordination stages of students at each grade level (5 th , 6 th , 7 th , and 8 th grades) were determined. The findings revealed that almost half of the middle school students operated at the lowest stage of unit coordination (Stage 1). While, at 5 th , 6 th , and 7 th grades, the number of students operating at Stage 1 was higher than at other unit coordination stages, there were more students at Stage 2 in 8 th grade. The number of students operating at Stage 3 was the lowest both at each grade level and in the whole group of 139 middle school students. Hence, very few of the middle school students could operate with three-level units as given and flexibly change between the three-level unit structures. To sum up, the results showed that middle school students could hardly work with and coordinate more than one-level units in mathematics problems requiring multiplicative reasoning. Considering these findings, we suggest conducting more studies examining the unit coordination stages of students and the ways for increasing students' unit coordination levels.

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“Approximate” multiplicative relationships between quantitative unknowns

Cited by 15 publications

References 33 publications

Teaching practices for differentiating mathematics instruction for middle school students

Teaching practices for differentiating mathematics instruction for middle school students

Causal pathways from fractions to algebra: Integrating psychology and math education perspectives

Investigation of middle school students’ unit coordination levels in mathematics problems involving multiplicative relations

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