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.

### Maria Blanton, Ana Stephens, Eric Knuth, Angela Murphy Gardiner, Isil Isler and Jee-Seon Kim

### Valerie N. Faulkner, Lee V. Stiff, Patricia L. Marshall, John Nietfeld and Cathy L. Crossland

This study is a longitudinal look at the different mathematics placement profiles of Black students and White students from late elementary school through 8th grade. Results revealed that Black students had reduced odds of being placed in algebra by the time they entered 8th grade even after controlling for performance in mathematics. An important implication of this study is that placement recommendations must be monitored to ensure that high-achieving students are placed appropriately, regardless of racial background.

### David Kirshner and Thomas Awtry

Information processing researchers have assumed that algebra symbol skills depend on mastery of the abstract rules presented in the curriculum (Matz, 1980; Sleeman, 1986). Thus, students' ubiquitous algebra errors have been taken as indicating the need to embed algebra in rich contextual settings (Kaput, 1995; National Council of Teachers of Mathematics [NCTM] Algebra Working Group, 1998). This study explored a nonrepresentational account of symbolic algebra skills as feature correlation within the visual field. We present evidence that algebra students respond spontaneously to the visual patterns of the notational display apart from engagement with the declarative content of the rules. Thus, persistent algebra errors may reflect disengagement from declarative content rather than an inability to deal with it. We sketch a Lexical Support System designed to sustain students' engagement with the declarative content of algebraic rules and processes, thus complementing the exciting curricular possibilities being developed for referentially rich algebra.

### David W. Carraher, Analúcia D. Schliemann, Bárbara M. Brizuela and Darrell Earnest

Algebra instruction has traditionally been postponed until adolescence because of historical reasons (algebra emerged relatively recently), assumptions about psychological development (“developmental constraints” and “developmental readiness”), and data documenting the difficulties that adolescents have with algebra. Here we provide evidence that young students, aged 9–10 years, can make use of algebraic ideas and representations typically absent from the early mathematics curriculum and thought to be beyond students' reach. The data come from a 30-month longitudinal classroom study of four classrooms in a public school in Massachusetts, with students between Grades 2–4. The data help clarify the conditions under which young students can integrate algebraic concepts and representations into their thinking. It is hoped that the present findings, along with those emerging from other research groups, will provide a research basis for integrating algebra into early mathematics education.

### Angeliki Kolovou, Marja van den Heuvel-Panhuizen and Olaf Köller

This study investigated whether an intervention including an online game contributed to 236 Grade 6 students' performance in early algebra, that is, solving problems with covarying quantities. An exploratory quasi-experimental study was conducted with a pretest-posttest-control-group design. Students in the experimental group were asked to solve at home a number of problems by playing an online game. Although boys outperformed girls in early algebra performance on the pretest as well as on the posttest, boys and girls profited equally from the intervention. Implications of these results for educational practice are discussed.

### Lulu Healy and Celia Hoyles

After surveying high-attaining 14-and 15-year-old students about proof in algebra, we found that students simultaneously held 2 different conceptions of proof: those about arguments they considered would receive the best mark and those about arguments they would adopt for themselves. In the former category, algebraic arguments were popular. In the latter, students preferred arguments that they could evaluate and that they found convincing and explanatory, preferences that excluded algebra. Empirical argument predominated in students' own proof constructions, although most students were aware of its limitations. The most successful students presented proofs in everyday language, not using algebra. Students' responses were influenced mainly by their mathematical competence but also by curricular factors, their views of proof, and their genders.

### Shulamit Kapon, Angela Halloun and Michal Tabach

We compared students' learning gains in authentic seventh-grade classrooms (N = 144) in 4 different interventions that incorporated a computer game that aims to teach players to solve linear equations. Significantly higher learning gains were measured in the implementations that were specifically designed to mediate the attribution of algebraic meaning to objects, actions, and rules in the game by engaging students in analogical mapping between these constructs and their algebraic counterparts and an exploration of the boundaries of this isomorphism. These findings suggest that learning disciplinary content and skills from a digital game requires learners to attribute disciplinary meaning to objects, actions, and rules in the game. Moreover, this process does not necessarily occur spontaneously and benefits from instructional mediation.

### Milan F. Sherman, Candace Walkington and Elizabeth Howell

Recent reform movements have emphasized students making meaning of algebraic relationships; however, research on student thinking and learning often remains disconnected from the design of widely used curricular materials. Although a previous examination of algebra textbooks (Nathan, Long, & Alibali, 2002) demonstrated a preference for a symbols-first approach, research has demonstrated that Algebra I students' performance on verbally presented problems is better than on symbolic equations, consistent with cognitive theories suggesting the value of concreteness fading. The present study investigates whether current textbooks used in Algebra I courses demonstrate a formalisms-first approach using five different analyses. Results show that despite nearly 2 decades of research on student learning, the conventional textbooks used in most classrooms have been resistant to change and emphasize manipulation with symbols prior to making sense of verbal scenarios.

### Maria L. Blanton and James J. Kaput

We present here results of a case study examining the classroom practice of one thirdgrade teacher as she participated in a long-term professional development project led by the authors. Our goal was to explore in what ways and to what extent the teacher was able to build a classroom that supported the development of students' algebraic reasoning skills. We analyzed 1 year of her classroom instruction to determine the robustness with which she integrated algebraic reasoning into the regular course of daily instruction and its subsequent impact on students' ability to reason algebraically. We took the diversity of types of algebraic reasoning, their frequency and form of integration, and techniques of instructional practice that supported students' algebraic reasoning as a measure of the robustness of her capacity to build algebraic reasoning. Results indicate that the teacher was able to integrate algebraic reasoning into instruction in planned and spontaneous ways that led to positive shifts in students' algebraic reasoning skills.

### Mollie MacGregor and Elizabeth Price

We have attempted to investigate whether 3 cognitive components of language proficiency—metalinguistic awareness of symbol, syntax, and ambiguity—are associated with students' success in learning the notation of algebra. Pencil-and-paper tests were given to assess students' metalinguistic awareness and their ability to use algebraic notation. In a total sample of more than 1500 students, aged 11 to 15, who were in their 1st to 4th years of algebra learning, we found that very few students with low metalinguistic awareness scores achieved high algebra scores. We discuss implications of this finding for the school algebra curriculum.