Making connections during math instruction is a recommended practice, but may increase the difficulty of the lesson. We used an avatar video instructor to qualitatively examine the role of linking multiple representations for 24 middle school students learning algebra. Students were taught how to solve polynomial multiplication problems, such as…

Mathematics curricula are frequently rich with visuals, but these visuals are often not designed for optimal use of students' limited cognitive resources. The authors of this study revised the visuals in a mathematics lesson based on instructional design principles. The purpose of this study is to examine the effects of these revised visuals on…

Solving mathematics story problems requires text comprehension skills. However, previous studies have found few connections between traditional measures of text readability and performance on story problems. We hypothesized that recently developed measures of readability and topic incidence measured by text-mining tools may illuminate associations…

This article explores the complementary strengths and weaknesses of grounded and abstract representations in the domain of early algebra. Abstract representations, such as algebraic symbols, are concise and easy to manipulate but are distanced from any physical referents. Grounded representations, such as verbal descriptions of situations, are…

Using the perspective of instructional conversation, we investigated how one teacher regulated student participation and conceptual reasoning in the middle-school mathematics classroom. We examined the elicitations--questions and provocative statements--made by the teacher over a four-day algebra lesson. Analyses showed how the teacher…

Algebraic reasoning stands as a formidable gatekeeper for students in their efforts to progress in mathematics and science, and to obtain economic opportunities (Ladson-Billings, 1998; RAND, 2003). Currently, mathematics education research has focused on algebra in order to provide access and opportunities for more students. There is now a growing…

Examines teachers' judgments of the difficulties of algebra problems. Offers classroom-tested instructional approaches that build directly on students' mathematical intuitions and inventions. (KHR)

Examined beliefs of elementary, middle, and high school mathematics teachers regarding algebra development. Found that teachers held a symbol-precedence view of student mathematical development. High school teachers were most likely to hold the symbol-precedence view and made the poorest predictions of students' performance, whereas middle school…

How the self-explanation process relates to learning and subsequent problem-solving performance was studied in two experiments with college students to examine whether students taught to self-explain during a study phase show greater test gains than control group counterparts; this is an attempt to replicate the results of M. T. H. Chi and others.…

This article explores how differences in problem representations change both the performance and underlying cognitive processes of beginning algebra students engaged in quantitative reasoning. Contrary to beliefs held by practitioners and researchers in mathematics education, students were more successful solving simple algebra story problems than…

Presents research in which mathematics teachers and educational researchers ordered arithmetic and algebra problems according to their predicted problem-solving difficulty for students. Analysis of students' problem-solving strategies indicates specific ways that students' algebraic reasoning differs from that predicted by most teachers and…

This paper presents a developmental model of students' acquisition of competence in quantitative and algebraic problem solving. A key notion underlying the developmental model is a distinction between grounded and abstract representations. Grounded representations, like story problems, are more concrete and familiar, closer to physical objects and…

This study (N = 48) examined the relationship between preservice secondary teachers' subject-matter expertise in mathematics and their judgments of students' algebra problem-solving difficulty. As predicted by the "expert blind spot" hypothesis, participants with more advanced mathematics education, regardless of their program affiliation or…

This study investigated middle school students' conceptual understanding of algebraic equations. 257 sixth- and seventh-grade students solved algebraic equations and generated story problems to correspond with given equations. Aspects of the equations' structures, including number of operations and position of the unknown, influenced students'…

I explore a belief about learning and teaching that is commonly held in education and society at large that nonetheless is deeply flawed. The belief asserts that mastery of "formalisms"--specialized representations such as symbolic equations and diagrams with no inherent meaning except that which is established by convention--is prerequisite to…