Can offline metacognition enhance mathematical problem solving?

The effectiveness of a short metacognitive intervention combined with algorithmic cognitive instruction was assessed in an elementary school setting. Two hundred thirty-seven 3rd-grade children were randomly assigned to a 5-session metacognitive strategy instruction, an algorithmic direct cognitive instruction, a motivational program, a quantitative-relational condition, or a spelling condition. Children in the metacognitive program achieved significant gains in trained metacognitive skills compared with the 4 other conditions. Moreover, the children in the metacognitive program performed better on trained cognitive skills than children in the algorithmic condition, with a follow-up effect on domain-specific mathematics problem-solving knowledge. Despite the consistency of findings, no generalization effects were found on transfer of cognitive learning. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

A life skills approach to mathematics instruction: preparing students with learning disabilities for the real-life math demands of adulthood

Current mathematics instruction does not address the day-to-day needs of many students with learning disabilities. Although the vast majority of students with learning disabilities are not college bound, much of mathematics instruction provides college preparation. Too often, classes in mathematics ignore the skills needed in home and community and on the job. The present article examines the ways in which general mathematics instruction, focused on daily living skills, can easily be integrated into the classrooms of students with learning disabilities.     

Learning mathematics: A cognitive perspective.

During the past decade, rather than studying the outcomes of mathematics learning in experimentation with specific teaching strategies, cognitive psychology has been advancing understanding of the fundamental nature of mathematics learning. The promise of cognitive theories for instruction is illustrated by reviewing several studies on elementary mathematics. This research illuminates the formal structure of a mathematical procedure such as counting and the hierarchy of its subprocedures, the diagnosis of consistent errors in subtraction and decimals and the discovery of their underlying sources, the formulation of the role of schemata in executing arithmetic skills, and the comprehension of word problems. The development of mathematics skills is considered in terms of the distinction between procedural and propositional knowledge. Implications of a cognitively based understanding of mathematical learning for the effective design of instruction are discussed. (67 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Mathematics education and students with learning disabilities: introduction to the special series

The prevalence of students with mathematics learning disabilities has triggered an interest among special education researchers and practitioners in developing an understanding of the needs of this group of students, and in identifying effective instructional programming to foster their mathematical performance during the school years and into adulthood. Research into the characteristics of students with mathematics learning disabilities is being approached from different perspectives, including developmental, neurological and neuropsychological, and educational. This diversity helps us develop a broader understanding of students' learning needs and difficulties. Special education assessment practices encompass a variety of approaches, including norm-referenced, criterion-referenced, and nonstandardized procedures, depending on the specific assessment questions professionals seek to answer. Students' mathematical knowledge and conceptual understanding must be examined to determine their strengths and weaknesses, curriculum-based progress, and use of cognitive strategies to arrive at mathematical solutions. Research findings have identified empirically validated interventions for teaching mathematics curricula to students with mathematics learning disabilities. Research studies have been grounded in behavioral theory and cognitive psychology, with an emergent interest in the constructivist approach. Although research studies have focused primarily on computational performance, more work is being conducted in the areas of story-problem solving and technology. These areas as well as other math curricular skills require further study. Additionally, the needs of adults with math LD have spurred educators to examine the elementary and secondary math curricula and determine ways to infuse them with life skills instruction accordingly. As the field of mathematics special education continues to evolve, special educators must remain cognizant of the developments in and influences on the field of mathematics education. Reform efforts have shaped the field significantly since the 1950s, contributing to the curriculum offered in mathematics textbooks and the pedagogical practices taught in higher education courses. Mathematics educators continue to search for a better understanding of how children learn mathematics; this process is shaped by the prevailing theoretical orientations and research methodologies. This special series in mathematics special education provides readers with information about the characteristics of students with mathematics learning disabilities, assessment procedures, mathematics programming, teacher preparation, and future directions for the field. The series originated as a result of discussions with Dr. Lee Wiederholt and Dr. Judith K. Voress, who saw a need for the compilation of recent research and best practices in mathematics special education. I thank them for their support of and thoughtful insights about the development of this series. I also appreciate the support of Dr. George Hynd and his editorial assistant, Kathryn Black, in finalizing the details for publication. Finally, I am most appreciative of the authors' contributions to this series; their work continues to significantly influence the development of the field of mathematics special education and programming for students with mathematics learning disabilities.     

Contribution of equal-sign instruction beyond word-problem tutoring for third-grade students with mathematics difficulty.

Elementary school students often misinterpret the equal sign (=) as an operational rather than a relational symbol. Such misunderstanding is problematic because solving equations with missing numbers may be important for the development of higher order mathematics skills, including solving word problems. Research indicates equal-sign instruction can alter how typically developing students use the equal sign, but no study has been conducted on the effects of such instruction for students with mathematics difficulty (MD) or how equal-sign instruction contributes to word-problem-solving skill for students with or without MD. In the present study, the authors assessed the efficacy of equal-sign instruction within word-problem tutoring. Third-grade students with MD (n = 80) were assigned to word-problem tutoring, word-problem tutoring plus equal-sign instruction (combined) tutoring, or no-tutoring control. Combined tutoring produced greater improvement on equal sign tasks and open equations than did the other 2 conditions. On certain forms of word problems, combined tutoring, but not word-problem tutoring alone, produced more improvement than did the control condition. When compared at posttest with 3rd-grade students without MD on equal-sign tasks and open equations, only combined tutoring students with MD performed comparably. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Effects of team assisted individualization on the mathematics achievement of academically handicapped and nonhandicapped students.

Evaluated the achievement effects of an individualized mathematics program designed to solve problems of management, direct instruction, and incentives common to earlier attempts at individualization of instruction. Team Assisted Individualization (TAI), a mathematics program that combines individualized instruction, cooperative learning teams, and direct instruction, was compared to control methods in a 24-wk experiment involving 1,371 students in 59 3rd-, 4th-, and 5th-grade classrooms. Standardized mathematics computations and concepts and application scales served as dependent variables and covariates. Random-effects nested analyses of covariance indicated statistically significant treatment effects favoring TAI for mathematics computations but not concepts and applications. However, individual-level analyses found significant treatment effects for both variables for the full sample and for a subsample of 63 academically handicapped Ss. No Treatment by Handicap interactions were found. (26 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Learning what's taught: Sex differences in instruction.

Research indicates that boys perform better on mathematics tests and girls perform better on reading tests. An investigation of why boys' and girls' performance differs was made by coding 33 teacher interactions with 2nd grade students during reading and mathematics instruction. Teachers made more academic contacts with girls in reading and with boys in math; teachers spent relatively more cognitive time with girls in reading and boys in math; teachers made consistently more managerial contacts with boys than girls; and, although there were no differences in initial abilities, sex differences were found in end-of-year achievement in reading. (35 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Fluency, accuracy, and gender predict developmental trajectories of arithmetic strategies.

The purpose of this study was to determine whether there are different growth trajectories of arithmetic strategies and whether these trajectories result in different achievement outcomes. Longitudinal data were collected on 240 students who began the study as 2nd graders. In the 1st year of the study, the 2nd-grade students were assessed on fluency and accuracy on simple arithmetic problems. During the fall of the 2nd, 3rd, and 4th grades, they were assessed on strategies for complex (multiple-digit) arithmetic. They were assessed on mathematics competency at the end of 4th grade. Growth mixture modeling was used to assess developmental trajectories in arithmetic strategies; the roles of fluency, accuracy, and gender in the development of latent class; and the impact of latent class on 4th-grade mathematics competency. The data indicated 2 latent classes of growth trajectories for correct cognitive strategy use and for attempted manipulative strategy use. Three latent classes were needed to explain the development of attempted cognitive strategy use. Fluency appeared to have the most significant impact on the growth rate, whereas accuracy and gender tended to influence the initial level of performance. Not all children transitioned away from manipulatives easily. A small latent class of children slightly increased their use of manipulatives over the course of the study, with the girls in this group being less likely than boys to abandon manipulatives. Finally, what appears to be the normal developmental trajectory for strategies was not found to serve many children well in regard to later mathematics achievement. (PsycINFO Database Record (c) 2011 APA, all rights reserved)     

Cognitive benefits and costs of bilingualism in elementary school students: The case of mathematical word problems.

Previous research has emphasized the importance of language for learning mathematics. This is especially true when mathematical problems have to be extracted from a meaningful context, as in arithmetic word problems. Bilingual learners with a low command of the instructional language thus may face challenges when dealing with mathematical concepts. At the same time, speaking two languages can be associated with cognitive benefits with regard to attentional control processes, although such benefits have only been found in highly proficient bilinguals. In the present study, we attempted to disentangle the effects of bilingual proficiency on mathematical problem solving in Turkish–German bilingual elementary school students. We examined whether the positive cognitive effects of bilingualism could be found not only in highly proficient bilinguals but also in students with an immigrant background and a low command of the instructional or native language. Our findings emphasize the importance of language proficiency for mathematics problem solving, as shown by the predictive value of students' proficiency in the language of testing (German/Turkish) for their performance on mathematical word problems. No additional effect of the language of instruction (German) was found for problem solving in the bilingual students' native language (Turkish). Furthermore, bilinguals gained scores comparable to those of their monolingual peers on word problems that required attentional control skills although performing significantly below their monolingual classmates on ordinary word problems, suggesting that bilinguals have an advantage when it comes to attentional control. Finally, bilingual students with a relatively high command of the instructional language performed better on word problems presented in German than on those presented in Turkish, thus facing cognitive costs when transferring knowledge from one language to the other. Implications of our findings for bilingual education are discussed. (PsycINFO Database Record (c) 2011 APA, all rights reserved)     

Examining students' retrospective memories of strategy development.

Two studies were conducted to assess students' self-reported use of cognitive study strategies. Both studies used a written survey to assess where and when students were introduced to their preferred study strategies. The 1st study compared high school and university students. The 2nd study compared university students' recollections of strategy use in high school with their present study behaviors. Results from the 2 studies were consistent. All groups, at all times in their education, regardless of course difficulty, preferred repetition strategies. Students were aware of other more sophisticated strategies but used them to a lesser extent. What was most compelling was students' indication that strategy instruction was a late addition to their educational experience, with many indicating high school as their 1st encounter with strategy instruction. These reports suggest that students' dependence on repetition strategies may be a product of the lack of familiarity and flexibility in the execution of more sophisticated strategies. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Teachers' knowledge of students' knowledge in mathematics problem solving: Correlational and case analyses.

This study examined the relationship of teachers' knowledge of students' knowledge to teachers' mathematics instruction and to students' mathematics problem solving. First-grade teachers (N?=?20) participated in a 4-week workshop in which they were given access to research-based knowledge on children's mathematics learning. Teachers were observed for 16 days throughout the school year. In May, teachers completed interviews and questionnaires about their knowledge and beliefs; their students completed achievement tests. Correlational analyses showed significant positive relationships between teachers' knowledge of students' knowledge and students' mathematics problem-solving achievement. Teachers with more knowledge of their students questioned students about problem-solving processes and listened to their responses. Teachers with less knowledge of their students explained problem-solving processes to students or observed students' solutions. Case analyses of knowledge and behavior of the most effective teacher and the least effective teacher supported these conclusions and showed important differences in how these teachers thought about and used students' knowledge. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

A comparison of the geometric reasoning of students attending Israeli ultraorthodox and mainstream schools.

Effects of schooling on a geometric misconception were examined by comparing the performance of Israeli students attending ultraorthodox schools with that of peers attending mainstream schools. These groups were of special interest because both value education highly and send essentially all children to school, but 1 group receives extensive instruction in mathematics and science and the other receives almost none. Despite the ultraorthodox 12–14 yr olds' having received no instruction in geometry, they more often solved the geometric misconception problems than did mainstream peers who had received extensive instruction in the subject. Mainstream 16–18 yr olds did somewhat better on the misconception task than did orthodox age peers, but even there, the advantage of the mainstream students was limited to those exposed to the most advanced mathematics curriculum. How mainstream and orthodox schooling may have contributed to these findings is discussed. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Dynamic assessment of algebraic learning in predicting third graders' development of mathematical problem solving.

Dynamic assessment (DA) involves helping students learn a task and indexing responsiveness to that instruction as a measure of learning potential. The purpose of this study was to explore the utility of a DA of algebraic learning in predicting third graders' development of mathematics problem solving. In the fall, 122 third-grade students were assessed on language, nonverbal reasoning, attentive behavior, calculations, word-problem skill, and DA. On the basis of random assignment, students received 16 weeks of validated instruction on word problems or received 16 weeks of conventional instruction on word problems. Then, students were assessed on word-problem measures proximal and distal to instruction. Structural equation measurement models showed that DA measured a distinct dimension of pretreatment ability and that proximal and distal word-problem measures were needed to account for outcome. Structural equation modeling showed that instruction (conventional vs. validated) and pretreatment calculation skills were sufficient to account for math word-problem outcome proximal to instruction; by contrast, language, pretreatment word-problem skill, and DA were needed to forecast learning on word-problem outcomes more distal to instruction. Findings are discussed in terms of responsiveness-to-intervention models for preventing and identifying learning disabilities. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Increasing Reading Comprehension and Engagement Through Concept-Oriented Reading Instruction.

Based on an engagement perspective of reading development, we investigated the extent to which an instructional framework of combining motivation support and strategy instruction (Concept-Oriented Reading Instruction--CORI) influenced reading outcomes for third-grade children. In CORI, five motivational practices were integrated with six cognitive strategies for reading comprehension. In the first study, we compared this framework to an instructional framework emphasizing Strategy Instruction (SI), but not including motivation support. In the second study, we compared CORI to SI and to a traditional instruction group (TI), and used additional measures of major constructs. In both studies, class-level analyses showed that students in CORI classrooms were higher than SI and/or TI students on measures of reading comprehension, reading motivation, and reading strategies. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

A quasi-experimental validation of transactional strategies instruction with low-achieving second-grade readers.

Second-grade, low-achieving students experienced a year of either transactional strategies instruction or highly regarded, more conventional second-grade reading instruction. By the end of the academic year, there was clear evidence of greater strategy awareness and strategy use, greater acquisition of information from material read in reading group, and superior performance on standardized reading tests by the transactional strategies instruction students. This is the clearest validation to date of educator-developed transactional strategies instruction. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Does concept-oriented reading instruction increase strategy use and conceptual learning from text?

The authors define reading engagement as the mutual support of motivations, strategies, and conceptual knowledge during reading. To increase reading engagement, a collaborative team designed a year-long integration of reading/language arts and science instruction (Concept-Oriented Reading Instruction, CORI). The authors compared students who received this instruction to similar students who received traditionally organized instruction aimed toward the same objectives. A path analysis showed that CORI had a positive effect on strategy use and text comprehension for students at Grades 3 and 5 when accounting for past achievement and prior knowledge. CORI also had a positive, indirect effect on conceptual knowledge mediated by strategy use, and this instruction facilitated conceptual transfer indirectly through several paths simultaneously. The findings are discussed in relation to a growing literature on instructional contexts for motivated strategy use and conceptual learning from text. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Mathematics Anxiety Rating Scale: Predicting anxiety experiences and academic performance in two groups of students.

Administered the Mathematics Anxiety Rating Scale (MARS) to 52 psychology students and 54 mathematics students. Preceding course examinations in statistics and mathematics courses, worry (cognitive concern about test performance), and emotionality (physiological and affective arousal) were assessed. MARS scores were found to be higher for psychology students than for mathematics students, to be useful predictors of both worry and emotionality, and to be inversely related to performance for psychology students. A strong inverse relationship was found between worry and performance for both groups and between emotionality and performance for psychology students. (25 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Alternative student evaluation structures and a focused schedule of instruction in an innercity junior high school.

Examined the effects of a particular model of direct instruction, focused instruction (FI), and 2 alternative student evaluation structures—individual learning expectations (ILE) and relative standing (RS)—on students' mathematics achievement and attitudes. Ss were 307 7th-grade students in 10 classes, who studied mathematics for 11 wks. Results indicated that FI, along with certificates of recognition, whether awarded on the basis of ILE or of RS, caused significantly higher achievement. There were no significant differences between the ILE and RS evaluation methods. (8 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

The impact of a strategy-based intervention on the comprehension and strategy use of struggling adolescent readers.

This study examines the impact of the Learning Strategies Curriculum (LSC), an adolescent reading intervention program, on 6th- and 9th-grade students' reading comprehension and strategy use. Using a randomized treatment–control group design, the study compared student outcomes for these constructs for 365 students who received daily instruction in 6 LSC strategies and 290 students who did not receive intervention instruction. After 1 school year, 6th-grade students who received intervention instruction significantly outperformed students in the control group on a standardized measure of reading comprehension and reported using problem-solving strategies in reading to a greater extent than students in the control group. There were no significant differences between 9th grade intervention and control groups in reading comprehension or strategy use. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Developmental differences in study behavior.

This study is a developmental examination of strategy instruction and the context that promotes when and how strategy instruction is maximized. The 4 experimental manipulations included comparisons between 2 strategy conditions, familiar and unfamiliar text, dyad versus individual study, and 4 age groups (M?=?10.5, 14.7, 19.9, and 21.9 yrs). The 486 students from Grades 5–6, and 9–10 and 1st- and 4th-yr university were assigned randomly to 1 strategy condition (self-study or elaborative interrogation) and 1 study context (dyad or individual). Participants studied and recalled 60 facts about familiar and unfamiliar animals. Explicit instruction in elaborative interrogation promoted memory performance, especially in the younger population. Studying in dyads enhanced memory and quality of study across age. To enhance text learning performance, students should be given explicit strategy instruction and should study interactively with their peers. (PsycINFO Database Record (c) 2010 APA, all rights reserved)