Search

This research focuses on the retention of students’ algebraic understandings 1 year following a 3-year early algebra intervention. Participants included 1,455 Grade 6 students who had taken part in a cluster randomized trial in Grades 3–5. The results show that, as was the case at the end of Grades 3, 4, and 5, treatment students significantly outperformed control students at the end of Grade 6 on a written assessment of algebraic understanding. However, treatment students experienced a significant decline and control students a significant increase in performance relative to their respective performance at the end of Grade 5. An item-by-item analysis performed within condition revealed the areas in which students in the two groups experienced a change in performance.

This research focuses on ways in which balance scales mediate students’ relational understandings of the equal sign. Participants included 21 Kindergarten–Grade 2 students who took part in an early...

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.

Guided by a theoretical framework emphasizing the importance of fidelity of implementation (FOI), this paper explores how teachers of grades 3, 4 and 5 implemented an early algebra interven...

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.

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...

In this article we advance characterizations of and supports for elementary students’ progress in generalizing and representing functional relationships as part of a comprehensive approach to early algebra. Our learning progressions approach to early algebra research involves the coordination of a curricular framework and progression, an instructional sequence, written assessments, and levels of sophistication describing students’ algebraic thinking. After detailing this approach, we focus on what we have learned about the development of students’ abilities to generalize and represent functional relationships in a grades 3–5 early algebra intervention by sharing the levels of responses we observed in students’ written work over time. We found that the sophistication of students’ responses increased over the course of the intervention from recursive patterning to correspondence and in some cases covariation relationships between variables. Students’ responses at times differed by the particular tas...

Research tells us that success in algebra is a factor in many other important student outcomes. Emerging research also suggests that students who are started on an algebra curriculum in the earlier grades may have greater success in the subject in secondary school. What’s needed is a consistent, algebra-infused mathematics curriculum all throughout the earlier grades. The authors outline three foundational principles for students to learn in the early grades, which can help them with the algebra they encounter in secondary school.

Researchers find that these classroom activities and instructional strategies support the development of third-grade students' algebraic thinking.

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.

In this chapter , we discus s the algebra framework that guides our work and how this framework was enacted in the design of a curricular approach for systematically developing elementary-aged students’ algebraic thinking. We provide evidence that, using this approach, students in elementary grades can engage in sophisticated practices of algebraic thinking based on generalizing, representing, justifying, and reasoning with mathematical structure and relationships. Moreover, they can engage in these practices across a broad set of content areas involving generalized arithmetic; concepts associated with equivalence, expressions, equations, and inequalities; and functional thinking.

This paper reports results from a written assessment given to 290 third-, fourth-, and fifth-grade students prior to any instructional intervention. We share and discuss students’ responses to items addressing their understanding of equation structure and the meaning of the equal sign. We found that many students held an operational conception of the equal sign and had difficulty recognizing underlying structure in arithmetic equations. Some students, however, were able to recognize underlying structure on particular tasks. Our findings can inform early algebra efforts by highlighting the prevalence of the operational view and by identifying tasks that have the potential to help students begin to think about equations in a structural way at the very beginning of their early algebra experiences.

This study examined how literal symbols affect students' understanding of algebraic expressions. Middle school students (N = 322) were randomly assigned to 1 of 3 conditions in which they were asked to interpret an expression (e.g., 4c + 3b) in a story problem. Each literal symbol represented the price of an item. In the c-and-b condition, the symbols used were the 1st letters of the items (e.g., price of a cake in dollars = c; price of a brownie in dollars = b). In the other 2 conditions, c and b were replaced with nonmnemonic English letters (x and y) or Greek letters (? and ?). Incorrect interpretations of the expression were most common among students in the c-and-b condition. Moreover, students in this condition were more likely than students in the other conditions to misinterpret the symbols as labels for objects (e.g., c stands for cake). An analysis of participating students' textbooks revealed that mnemonic symbols were used correctly and were not uncommon. Results suggest that the use of mnemonic symbols may hinder students' interpretation of algebraic expressions.

Middle school students with learning disabilities in math (MLD) used two versions of Enhanced Anchored Instruction (EAI). In one condition, students learned how to compute with fractions on an as-needed basis while they worked to solve the EAI problems. In the other condition, teachers used a computer-based instructional module in place of one of the EAI problems to deliver formal fraction instruction. The results indicated that students in both instructional formats improved their fraction computational skills and that formal instruction provided an added benefit. Both instructional conditions improved students' problem-solving skills by about the same amount. The findings suggest that combining formal fraction instruction with EAI is a viable way to improve the problem-solving and computational skills of students with MLD.

While curriculum specialists and committees often decide how mathematics is taught, it is ultimately principals who influence the extent to which these initiatives are carried out. The overall goal of this article is to provide school leaders with classroom-based research that describes one way of improving the math skills of middle school students. The study employed a randomized pretest-posttest comparison group design to examine the effects of two versions of Enhanced Anchored Instruction (EAI) and a Business as Usual (BAU) condition on the math skills of middle school students in technology education classrooms. Results showed that both EAI conditions were effective at improving the math skills of students over those of students in the BAU classes. The findings suggest that technology education teachers can make important contributions in helping students develop their computation and problem-solving skills.

Building a skateboard ramp motivates students to practice and apply fraction operations and measurement while developing problem-solving skills.

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).

This study examined conceptions of algebra held by 30 preservice elementary teachers. In addition to exploring participants’ general “definitions” of algebra, this study examined, in particular, their analyses of tasks designed to engage students in relational thinking or a deep understanding of the equal sign as well as student work on these tasks. Findings from this study suggest that preservice elementary teachers’ conceptions of algebra as subject matter are rather narrow. Most preservice teachers equated algebra with the manipulation of symbols. Very few identified other forms of reasoning – in particular, relational thinking – with the algebra label. Several participants made comments implying that student strategies that demonstrate traditional symbol manipulation might be valued more than those that demonstrate relational thinking, suggesting that what is viewed as algebra is what will be valued in the classroom. This possibility, along with implications for mathematics teacher education, will be discussed.

This article reports results from a study focused on teachers' knowledge of students' understanding of core algebraic concepts. In particular, the study examined middle school mathematics teachers' knowledge of students' understanding of the equal sign and variable, and students' success applying their understanding of these concepts. Interview data were collected from 20 middle school teachers regarding their predictions of student responses to written assessment items focusing on the equal sign and variable. Teachers' predictions of students' understanding of variable aligned to a large extent with students' actual responses to corresponding items. In contrast, teachers' predictions of students' understanding of the equal sign did not correspond with actual student responses. Further, teachers rarely identified misconceptions about either variable or the equal sign as an obstacle to solving problems that required application of these concepts. Implications for teacher professional development are discussed.

This longitudinal study investigated (a) middle school students' understanding of the equal sign, (b) students' performance solving equivalent equations problems, and (c) changes in students' understanding and performance over time. Written assessment data were collected from 81 students at four time points over a 3-year period. At the group level, understanding and performance improved over the middle school years. However, such improvements were gradual, with many students still showing weak understanding and poor performance at the end of grade 8. More sophisticated understanding of the equal sign was associated with better performance on equivalent equations problems. At the individual level, students displayed a variety of trajectories over the middle school years in their understanding of the equal sign and in their performance on equivalent equations problems. Further, students' performance on the equivalent equations problems varied as a function of when they acquired a sophisticated understanding...

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).

This study examined how 4 middle school textbook series (2 skills-based, 2 Standards-based) present equal signs. Equal signs were often presented in standard operations equals answer contexts (e.g., 3 + 4 = 7) and were rarely presented in nonstandard operations on both sides contexts (e.g., 3 + 4 = 5 + 2). They were, however, presented in other nonstandard contexts (e.g., 7 = 7). Two follow-up experiments showed that students' interpretations of the equal sign depend on the context. The other nonstandard contexts were better than the operations equals answer context at eliciting a relational understanding of the equal sign, but the operations on both sides context was best. Results suggest that textbooks rarely present equal signs in contexts most likely to elicit a relational interpretation—an interpretation critical to success in algebra.

This study examined awareness of equivalence and relational thinking exhibited by 30 preservice elementary teachers in order to assess their initial preparedness to engage students in these two important aspects of early algebraic reasoning. Findings indicated that preservice teachers collectively demonstrated an awareness of relational thinking both in identifying opportunities offered by tasks to engage students in this thinking and in identifying this thinking in samples of student work. However, in proposing difficulties students might have with selected tasks, few participants demonstrated the understanding that many elementary school students hold misconceptions about the meaning of the equal sign. Implications of these findings for preservice and inservice teacher education are discussed.

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.

Algebra is a focal point of reform efforts in mathematics education, with many mathematics educators advocating that algebraic reasoning should be integrated at all grade levels K-12. Recent research has begun to investigate algebra reform in the context of elementary school (grades K-5) mathematics, focusing in particular on the development of algebraic reasoning. Yet, to date, little research has focused on the development of algebraic reasoning in middle school (grades 6–8). This article focuses on middle school students’ understanding of two core algebraic ideas—equivalence and variable—and the relationship of their understanding to performance on problems that require use of these two ideas. The data suggest that students’ understanding of these core ideas influences their success in solving problems, the strategies they use in their solution processes, and the justifications they provide for their solutions. Implications for instruction and curricular design are discussed.