One important aspect of mathematical understanding is the ability to reason within, and translate among, multiple representations (Brenner et al. 1997; Moschkovich, Schoenfeld, and Arcavi 1993). The National Council of Teachers of Mathematics (2000) recognizes this aspect and calls for an increased focus on a variety of mathematical representations, including tables, graphs, algebraic expressions, and verbal expressions, as well as the interconnections among them. The informal study described in this article is specifically concerned with the connection between verbal and algebraic representations of mathematical situations.
Ana C. Stephens
Ana C. Stephens
Algebra's “Gatekeeper” status has prompted several in the mathematics education research community (e.g., Davis 1985; Kaput 1998; Olive, Izsak, and Blanton 2002) to urge educators to view algebra not as an isolated course but as a continuous K–12 strand that is present throughout the entire mathematics curriculum. Central to the transition from arithmetic to algebraic reasoning is the concept of variable (Schoenfeld and Arcavi 1988). Schoenfeld and Arcavi argue that despite its importance, most mathematics curricula offer little to assist students in developing ideas about this concept. They assert that instead of providing students opportunities to practice manipulating terms and solving for unknowns, teachers should encourage students to view variables as shorthand tools for expressing already-understood ideas about varying quantities. This article describes a mathematical problem that can encourage students to view variables in this way while confronting a common misconception.
Laura Grandau and Ana C. Stephens
Research on the learning and teaching of algebra has recently been identified as a priority by members of the mathematics education research community (e.g., Ball 2003; Carpenter and Levi 2000; Kaput 1998; Olive, Izsak, and Blanton 2002). Rather than view algebra as an isolated course of study to be completed in the eighth or ninth grade, these researchers advocate the reconceptualization of algebra as a strand that weaves throughout other areas of mathematics in the K–12 curriculum.
Ana C. Stephens and Crystal L. Lamers
Increased calls for student, teacher, and school “accountability” have placed a greater emphasis on the need to assess what students know. While the focus of the calls for accountability is often on high-stakes testing, there are other important forms and purposes of assessment that deserve attention. In Principles and Standards for School Mathematics, the National Council of Teachers of Mathematics (2000) asserts, “Assessment should be more than merely a test at the end of instruction to see how students perform under special conditions; rather, it should be an integral part of instruction that informs and guides teachers as they make instructional decisions. Assessment should not merely be done to students; rather, it should also be done for students, to guide and enhance their learning” (p. 22).
Ana Stephens, Maria Blanton, Eric Knuth, Isil Isler and Angela Murphy Gardiner
Researchers find that these classroom activities and instructional strategies support the development of third-grade students' algebraic thinking.
Isil Isler, Tim Marum, Ana Stephens, Maria Blanton, Eric Knuth and Angela Murphy Gardiner
Engage your students in functional thinking—an important precursor to algebra—with this classroom activity.
Eric J. Knuth, Ana C. Stephens, Nicole M. McNeil and Martha W. Alibali
Given its important role in mathematics as well as its role as a gatekeeper to future educational and employment opportunities, algebra has become a focal point of both reform and research efforts in mathematics education. Understanding and using algebra is dependent on understanding a number of fundamental concepts, one of which is the concept of equality. This article focuses on middle school students' understanding of the equal sign and its relation to performance solving algebraic equations. The data indicate that many students lack a sophisticated understanding of the equal sign and that their understanding of the equal sign is associated with performance on equation-solving items. Moreover, the latter finding holds even when controlling for mathematics ability (as measured by standardized achievement test scores). Implications for instruction and curricular design are discussed.
Eric J. Knuth, Martha W. Alibali, Shanta Hattikudur, Nicole M. McNeil and Ana C. Stephens
The equal sign is perhaps the most prevalent symbol in school mathematics, and developing an understanding of it has typically been considered mathematically straightforward. In fact, after its initial introduction during students' early elementary school education, little, if any, instructional time is explicitly spent on the concept in the later grades. Yet research suggests that many students at all grade levels have not developed adequate understandings of the meaning of the equal sign (Baroody and Ginsburg 1983; Behr, Erlwanger, and Nichols 1980; Falkner, Levi, and Carpenter 1999; Kieran 1981; Knuth et al. 2006). Such findings are troubling with respect to students' preparation for algebra, especially given Carpenter, Franke, and Levi's (2003) contention that a “limited conception of what the equal sign means is one of the major stumbling blocks in learning algebra. Virtually all manipulations on equations require understanding that the equal sign represents a relation” (p. 22).
Ana C. Stephens, Brian A. Bottge and Enrique Rueda
Building a skateboard ramp motivates students to practice and apply fraction operations and measurement while developing problem-solving skills.
Maria Blanton, Ana Stephens, Eric Knuth, Angela Murphy Gardiner, Isil Isler and Jee-Seon Kim
This article reports results from a study investigating the impact of a sustained, comprehensive early algebra intervention in third grade. Participants included 106 students; 39 received the early algebra intervention, and 67 received their district's regularly planned mathematics instruction. We 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. We found that the intervention group significantly outperformed the nonintervention group and was more apt by posttest to use algebraic strategies to solve problems. Given the multitude of studies among adolescents documenting students' difficulties with algebra and the serious consequences of these difficulties, an important contribution of this research is the finding that—provided the appropriate instruction—children are capable of engaging successfully with a broad and diverse set of big algebraic ideas.