This article introduces unit transformation graphs (UTGs) as a tool for diagramming the ways students use sequences of mental actions to solve mathematical tasks. We report findings from a study in which we identified patterns in the ways preservice elementary school teachers relied on working memory to coordinate mental actions when operating in…

Despite progress toward gender equity, troubling disparities in mathematical problem-solving performance and related outcomes persist. To investigate why, we build on recurrent findings in previous studies to introduce a new construct, "bold problem solving," which involves approaching mathematics problems in inventive ways. We introduce…

We examined geometric calculation with number tasks used within a unit of geometry instruction in a Taiwanese classroom, identifying the source of each task used in classroom instruction and analyzing the cognitive complexity of each task with respect to 2 distinct features: diagram complexity and problem-solving complexity. We found that…

This article centers on studying the progress of algebraic syntax once students have surmounted initial obstacles found in the transition toward symbolic algebra. It specifically analyzes students' progress concerning the operation on the unknown, when the latter is represented by an expression that involves a 2nd unknown. In curricular terms,…

Sets of mathematics problems are generally arranged in 1 of 2 ways. With "blocked practice," all problems are drawn from the preceding lesson. With "mixed review," students encounter a mixture of problems drawn from different lessons. Mixed review has 2 features that distinguish it from blocked practice: Practice problems on…

The problem-solving processes of seven upper-division college students working a series of problems that could be solved by the application of one or more heuristics are described in detail. (MP)

Explores insights that may be provided by a situated perspective on learning. Considers the ways in which a focus on the classroom community and the behaviors and practices implicit within such communities may increase understanding of students' mathematical knowledge production and use. (Contains 22 references.) (Author/ASK)

Forty second-year algebra students were interviewed while solving eight problems. Data from the interviews included frequencies of problem-solving processes and scores on the test. Conceptual knowledge accounted for 50 percent of the variance. (MP)

Approaches to attaining coder agreement when analyzing protocols of problem-solving processes are considered. A new procedure for estimating consensus is presented. (MNS)

The subjects were 47 first-grade children from three classrooms in a Lexington, Kentucky public school. Results indicate that the position of the unknown set in a verbal problem substantially determines whether or not a problem can be modeled successfully by children of this age. (MP)

Some easily graded measures of problem-solving processes are introduced, and the impact of a month-long intensive problem-solving course on a selected group of college freshmen and sophomores is demonstrated. The measures are thought to have shown themselves to be both reliable and informative. (MP)

This study investigated students' understanding of two mathematical relations as measured by conservation tasks. Thirty students were interviewed, half each at the middle and high school levels. The results suggest that the ability to conserve equation or function varies among pupils of different ages, sexes, and mathematical backgrounds. (MP)

Investigates the calculation strategies used by female students in Grades Two and Three to solve word problems. Findings indicate that students used three main intuitive models: (1) direct counting; (2) repeated addition; and (3) multiplicative operation. Concludes that children acquire an expanding repertoire of intuitive models and the model…

Included are a general introduction to the topic of metacognition, a discussion of how metacognition is involved in mathematical performance, and a short section on metacognition and instruction. (MNS)

Eight boys and eight girls from a rural elementary school participated in the investigation. Specific heuristics were adopted from Polya; and the students selected represented two substages of Piaget's concrete operational stage. Five hypotheses were generated, based on observed results and the study's theoretical rationale. (MP)