Research on the Learning and Teaching of Algebra

Cirino

et al. 2015

Exceptional Children

The focus of the present study was enhancing word-problem and calculation achievement in ways that support pre-algebraic thinking among 2nd-grade students at risk for mathematics difficulty. Intervention relied on a multi-tier support system (i.e., responsiveness-to-intervention or RTI) in which at-risk students participate in general classroom instruction and receive supplementary small-group tutoring. Participants were 265 students in 110 classrooms in 25 schools. Teachers were randomly assigned to 3 conditions: calculation RTI, word-problem RTI, and business-as-usual control. Intervention lasted 17 weeks. Multilevel modeling indicated that calculation RTI improved calculation but not word-problem outcomes; word-problem RTI enhanced proximal word-problem outcomes as well as performance on some calculation outcomes; and word-problem RTI provided a stronger route than calculation RTI to pre-algebraic knowledge.

Section: Pre-algebraic Thinking and Possible Connections To Calculatimentioning

confidence: 99%

Effects of a Multitier Support System on Calculation, Word Problem, and Prealgebraic Performance Among At-Risk Learners

Powell

Cirino

et al. 2015

Exceptional Children

“…For example, competent problem solvers translate “Fred had 3 more than Harry” to F = 3 + H, by recognizing the smaller quantity must increase to equal the larger quantity. A common error, however, is F + 3 = H. Word problems may, therefore, involve greater symbolic complexity than calculations and may rely more on the type of mental flexibility, manipulation of symbolic associations, and maintenance of multiple representations (numerical and linguistic) that support pre-algebraic thinking (e.g., Kieran, 1992; Sfard & Linchevski, 1994). …”

Section: The Role Of Arithmetic Calculations and Word Problems In Algmentioning

confidence: 99%

Does calculation or word-problem instruction provide a stronger route to prealgebraic knowledge?

Journal of Educational Psychology

Powell

Cirino

et al. 2014

The focus of this study was connections among 3 aspects of mathematical cognition at 2nd grade: calculations, word problems, and pre-algebraic knowledge. We extended the literature, which is dominated by correlational work, by examining whether intervention conducted on calculations or word problems contributes to improved performance in the other domain and whether intervention in either or both domains contributes to pre-algebraic knowledge. Participants were 1102 children in 127 2nd-grade classrooms in 25 schools. Teachers were randomly assigned to 3 conditions: calculation intervention, word-problem intervention, and business-as-usual control. Intervention, which lasted 17 weeks, was designed to provide research-based linkages between arithmetic calculations or arithmetic word problems (depending on condition) to pre-algebraic knowledge. Multilevel modeling suggested calculation intervention improved calculation but not word-problem outcomes; word-problem intervention enhanced word-problem but not calculation outcomes; and word-problem intervention provided a stronger route than calculation intervention to pre-algebraic knowledge.

“…In this vein, we use the term algebra in a limited way to mean representing and reasoning about problem situations that contain unknowns (Izsak, 2000) and solving problems using the kinds of mathematical expressions found in algebra (Kiernan, 1992). 1 For years, some (e.g., Davis, 1985, 1989; Kaput, 1995) have advocated incorporating algebra throughout the K–12 curriculum to add coherence and depth to school mathematics and to ease the transition to formal algebra.…”

Section: Why Incorporate Algebra To Represent the Underlying Structurmentioning

confidence: 99%

“…More recently, the Algebra Working Group (NCTM, 1997) further conceptualized an early focus on algebra, prompting the 2000 NCTM standards to encourage teachers to nurture students’ emerging knowledge of algebra beginning in kindergarten. As noted, in the present study we use algebra in a limited way: to represent and solve problems and to reason about relations among operations (e.g., Izsak, 2000; Kiernan, 1992). …”

Section: Why Incorporate Algebra To Represent the Underlying Structurmentioning

confidence: 99%

The Effects of Schema-Broadening Instruction on Second Graders' Word-Problem Performance and Their Ability to Represent Word Problems with Algebraic Equations: A Randomized Control Study

The Elementary School Journal

Zumeta

Schumacher

et al. 2010

The purpose of this study was to assess the effects of schema-broadening instruction (SBI) on second graders' word-problem-solving skills and their ability to represent the structure of word problems using algebraic equations. Teachers (n = 18) were randomly assigned to conventional word-problem instruction or SBI word-problem instruction, which taught students to represent the structural, defining features of word problems with overarching equations. Intervention lasted 16 weeks. We pretested and posttested 270 students on measures of word-problem skill; analyses that accounted for the nested structure of the data indicated superior word-problem learning for SBI students. Descriptive analyses of students' word-problem work indicated that SBI helped students represent the structure of word problems with algebraic equations, suggesting that SBI promoted this aspect of students' emerging algebraic reasoning.When solving word problems, students are faced with novel problems that require transfer. This can be difficult to effect in the primary grades (Durnin, Perrone, & MacKay, 1997; Foxman, Ruddock, McCallum, & Schagen, 1991, cited in Boaler, 1993Larkin, 1989). Some psychologists view such transfer in terms of the development of schemas, by which students conceptualize word problems within categories or problem types that share structural, defining features and require similar solution methods (Chi, Feltovich, & Glaser, 1981; Gick & Holyoake, 1983;Mayer, 1992;Quilici & Mayer, 1996). The broader the schema or problem type, the greater the probability students will recognize connections between novel problems and those used for instruction and will understand when to apply the solution methods they have learned.In a series of studies, we have relied on this conceptualization of transfer to design instruction for helping students build schemas for word-problem types and for broadening those schemas. Prior work (e.g., Fuchs et al., 2003;Fuchs, Fuchs, Prentice, et al., 2004) illustrates the efficacy of this approach, which we refer to as schema-broadening instruction (SBI), for third-grade students on problem types relevant to the third-grade curriculum. More recently, Fuchs, Powell et al. (2009) demonstrated the efficacy of SBI tutoring for a subset of third graders who experience severe mathematics difficulty. With this population, we taught simpler word-problem types, while introducing algebraic equations to represent the defining, structural features of those problem types.Inquiries should be sent to Lynn S. Fuchs, Box 328 Peabody, Vanderbilt University, Nashville,; lynn.fuchs@vanderbilt.edu). NIH Public Access Author ManuscriptElem Sch J. Author manuscript; available in PMC 2010 June 9. NIH-PA Author ManuscriptNIH-PA Author Manuscript NIH-PA Author ManuscriptWe extend this line of work in the present study, applying SBI to problem types appropriate to the second-grade curriculum. We focused on typically developing second-grade students while relying on a whole-class format to deliver instruction, again ...