The nature of the development of arithmetic performance has long been intensively studied, and available scientific evidence can be evaluated and synthesized in light of Nelson and Narens' model of metacognition. According to the Nelson-Narens model, human cognition can be split into two or more interrelated levels. Obviously, in the case of more…

Proficiency in mathematics across the United States has become an area of concern as performance on state accountability tests continues to demonstrate that less than 50% of students in 4th and 8th grade are proficient in mathematics (NCES, 2013). With such large numbers of students lacking proficiency in mathematics, systematic interventions that…

This study investigated the role of cognitive interventions on word problem solving (WPS) accuracy in children who are English learners (ELs) and at risk for math difficulties (MD). Grade 3 children (n = 158) within classrooms were randomly assigned to either an untreated control group or one of three treatment conditions: paraphrase question…

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…

Reports observations of the use of mental aerobics with 48 adults whose median age was 70. Provides examples of the group puzzles and logic, math, and word problems used to enhance cognitive functioning and creative thinking. (SK)

Tested three models of children's mathematics word-problem solving based on developmental differences in quantitative conceptual structures: (1) quantitative relations represented as ordered array of mental objects; (2) numbers represented on two tentatively coordinated mental number lines; and (3) numerical operations represented as objects on…

Devised a model that hypothesized that students, in discriminating relevant information in mathematical story problems, search for values of semantic categories that match those in the problem. Two experiments assessed the validity of the model. Results indicated that the model was an accurate predictor of discrimination performance for successful…

The influence of cognitive growth in working memory (WM) on mathematical problem solution accuracy was examined in elementary school children (N = 353) at risk and not at risk for serious math problem solving difficulties. A battery of tests was administered that assessed problem solving, achievement, and cognitive processing (WM, inhibition,…

Describes qualitative aspects of explicit and less explicit explanations of cognitive reasoning processes associated with mathematical problem solving strategies. Includes lesson excerpts to illustrate three major differences between explicit and less explicit explanations. (MG)

This study compared the performances of college mathematics students on mathematics problems stated in three different contexts: in arithmetical form, in the form of algebraic expressions to be simplified, and in the form of practical situations stated in words. Both single-step and multi-step problems were used. For all three contexts, there were…

This book shows how to increase one's power to analyze and comprehend problems. First, it outlines and illustrates the methods that good problem solvers use in attacking complex ideas. Then it gives some practice in applying these methods to a variety of questions in comprehension and reasoning. Chapters include: (1) "Test Your Mind--See How…

Measure Up is a research and development project that uses findings from Davydov (1975) and others to introduce mathematics through measurement and algebra in grades 1-3. This paper illustrates the use of generalized diagrams and symbols in solving word problems for a group of 10 children selected from a grade 3 Measure Up classroom. Students use…

Pragmatic and semantic problem solving are examined as processes that enhance acquisition of mathematical knowledge. It is suggested that development of mathematical cognition involves restructuring and that math teachers can help restructure children's knowledge systems by providing them with situations in which semantic and pragmatic problem…

This article describes a problem-solving strategy unit to be used as a supplement to the regular mathematics curriculum at the primary level. Specific teaching steps and examples are given for three developmental stages of thinking: (1) concrete, (2) representational, and (3) abstract. (DB)

A mapping procedure based on the SOLO Taxonomy developmental model was used to classify the problem-solving strategies of students (n=34) in grades K-2. Only one multiplication problem was used to isolate three components of the problem-solving procedure. (MDH)