Understanding and solving word arithmetic problems.

Presents a processing model that deals explicitly with both the text-comprehension and problem-solving aspects of word arithmetic problems. General principles from a theory of text processing developed by T. A. Van Dijk and the 1st author (1983) are combined with hypotheses by M. S. Riley et al (1983) about semantic knowledge for understanding problem texts in an integrated model of problem comprehension. The model simulates construction of cognitive representations that include information that is appropriate for problem-solving procedures that children use. Several information-processing steps are distinguished, and various levels of representation are described. The model provides an analysis of processing requirements, including requirements for short-term memory that differ among types of problems. Predictions about difficulty of problems based on these processing differences are generally consistent with reported data. (38 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Students' miscomprehension of relational statements in arithmetic word problems.

In Experiment 1, students were asked to write their complete solution processes for arithmetic word problems containing relational statements. Students were more likely to miscomprehend a relational statement when the required arithmetic operation was inconsistent with the statement's relational term, such as having to subtract when the relational term was more than. This effect was magnified when the relational term was marked (e.g., less than) rather than unmarked (e.g., more than). In Experiment 2, students were given information about two variables and asked to generate a statement expressing the relation between them. Students tended to produce relational statements by using unmarked rather than marked comparative terms. Finally, we present a model of word problem comprehension processes that uses schemata as guides to comprehension. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Solving arithmetic word problems: Role of reading and computational skills.

Investigated how children cope with some of the demands imposed on them by arithmetic word problems by administering problems modeled after those used by the National Assessment of Educational Progress to 200 6th-graders. A computational demand was imposed on the Ss by adding extraneous information to the problems, whereas a reading demand was imposed on them by increasing the syntactic complexity of the problems. Multiple regression analyses indicated that the Ss' computational ability and reading ability together accounted for 54% of the variance in solution accuracy: Eight and 14%, respectively, of this variance was unique, whereas 32% was common to the abilities. In addition, the analyses indicated that the presence of extraneous information in the problems reduced the accuracy of Ss' solutions. The use of complex syntax had no significant effect on accuracy. The findings suggest that reading ability and computational ability both play important roles in children's successful solution of word problems. The findings also suggest that the presence of extraneous information in word problems can impose a formidable demand on children's limited processing capacities. (14 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Comprehension of arithmetic word problems: Evidence from students' eye fixations.

Students have difficulty solving arithmetic word problems containing a relational term that is inconsistent with the required arithmetic operation (e.g., containing the term less, yet requiring addition) rather than consistent. To investigate this consistency effect, students' eye fixations were recorded as they read arithmetic word problems on a computer monitor and stated a solution plan for each problem. As predicted, low-accuracy students made more reversal errors on inconsistent than consistent problems, students took more time for inconsistent than consistent problems, this additional time was localized in the integration/planning stages of problem solving rather than in the initial reading of the problem, these response-time patterns were obtained for high-accuracy but not for low-accuracy students, and high-accuracy students required more rereadings of previously fixated words for inconsistent than for consistent problems. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

The cognitive correlates of third-grade skill in arithmetic, algorithmic computation, and arithmetic word problems.

The purpose of this study was to examine the cognitive correlates of 3rd-grade skill in arithmetic, algorithmic computation, and arithmetic word problems. Third graders (N = 312) were measured on language, nonverbal problem solving, concept formation, processing speed, long-term memory, working memory, phonological decoding, and sight word efficiency as well as on arithmetic, algorithmic computation, and arithmetic word problems. Teacher ratings of inattentive behavior also were collected. Path analysis indicated that arithmetic was linked to algorithmic computation and to arithmetic word problems and that inattentive behavior independently predicted all 3 aspects of mathematics performance. Other independent predictors of arithmetic were phonological decoding and processing speed. Other independent predictors of arithmetic word problems were nonverbal problem solving, concept formation, sight word efficiency, and language. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Role of examples in how students learn to categorize statistics word problems.

In Experiment 1, students who studied example word problems that were grouped by t test, correlation, and chi-square were more likely to sort subsequent problems on the basis of structure and less likely to sort on the basis of surface characteristics than students who received no examples. In Experiment 2, this pattern was strongest when students studied structure-emphasizing rather than surface-emphasizing examples. In Experiment 3, students who studied and practiced 4 structure-emphasizing worked-out examples of t test and correlation problems were more likely to apply the appropriate statistical test correctly to subsequently presented statistics word problems than students who had studied surface emphasizing examples. This pattern was strong for lower but not for higher ability students. Implications of a schema construction theory are discussed. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Comprehension of arithmetic word problems: A comparison of successful and unsuccessful problem solvers.

It is proposed that when solving an arithmetic word problem, unsuccessful problem solvers base their solution plan on numbers and keywords that they select from the problem (the direct translation strategy), whereas successful problem solvers construct a model of the situation described in the problem and base their solution plan on this model (the problem-model strategy). Evidence for this hypothesis was obtained in 2 experiments. In Experiment 1, the eye fixations of successful and unsuccessful problem solvers on words and numbers in the problem statement were compared. In Experiment 2, the degree to which successful and unsuccessful problem solvers remember the meaning and exact wording of word problems was examined. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Detecting unsolvable algebra word problems.

When do students detect algebra word problems with missing information, and how do they do it? To determine whether such detection operates automatically or requires conscious effort and attention, this study manipulated whether a hint was provided that problems might be unsolvable. A signal detection analysis revealed that the hint resulted in a large improvement in the detection of unsolvable problems, implying that many individuals possess the means to detect missing information but that conscious effort is required for those means to be deployed. Participants with moderate mathematical ability had good success at detecting missing information when given the hint, but only when the problems had familiar cover stories. High math ability individuals were successful regardless of the cover story's familiarity. In addition, participants were more likely to conclude that a solvable problem was unsolvable when the hint was provided, when the problems' cover story was unfamiliar, and with the passage of time. As a result, problem solving of solvable problems was sometimes abandoned prematurely. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

What makes certain arithmetic word problems involving the comparison of sets so difficult for children?

Arithmetic word problems with an unknown reference set, such as "John has 7 eggs. He has 4 eggs fewer [more] than Peter. How many eggs does Peter have?" are considerably more difficult for children than problems with an unknown compare set (2nd sentence: "Peter has 4 eggs more [fewer] than John"). Six experiments with 1st graders and kindergartners investigated reasons for this finding. Exps 1–4 revealed that neither difficulties in processing the personal pronoun nor the use of key word strategies could explain the difficulty differences. Exp 5 showed that most 1st graders were not aware that the difference between 2 sets can be expressed by either "In Set x there are n more objects than in Set y" or "In Set y there are n fewer objects than in Set x." Exp 6 indicated that this lack of access to flexible language use is what makes compare problems with an unknown reference so difficult. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

A structure-mapping model for word problems.

Tested predictions of a structure-mapping model for word problems in 4 experiments involving 132 undergraduates. In Exps I and II, Ss rated the potential usefulness of solutions for pairs of problems—mixture problems in Exp I and work problems in Exp II. The problems were either equivalent (same story, same procedure), similar (same story, different procedure), isomorphic (different story, same procedure), or unrelated (different story, different procedure). Ss in Exp III used an example solution for a work problem and a mixture problem to generate equations to related test problems that differed in their mappings from the example. In Exp IV, Ss matched concepts in the test problems to corresponding concepts in the examples to provide a direct measure of their ability to construct mappings across different problems. In Exps III and IV, Ss performed significantly better on isomorphic problems than on similar problems, and significantly better on work isomorphs than on mixture isomorphs. Results suggest that a structure-mapping model that emphasizes the transparency and structure of the mapping can be used to predict the usefulness of a solution. (19 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Selecting examples for solving word problems.

College students studied the solution of a simple and a complex example, then they used the examples to construct equations for algebra word problems. The frequency with which they referred to each example across 8 test problems that varied in complexity was recorded. In a 5-factor ANOVA, instructional format, test-problem order, and aptitude were between-Ss factors, and example complexity and test-problem complexity were within-S factors. Evidence for good metacognitive skills regarding selection of examples included students' ability to match examples with test problems and their preference for the complex example. However, neither finding interacted with instructional format or aptitude. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Effect of computer graphics on improving estimates to algebra word problems.

Evaluated the effectiveness of computer graphics in improving estimates for algebra word problems in 4 experiments involving 180 undergraduates. Exp I attempted to improve estimates for average-speed problems, Exp II attempted to improve estimates for tank problems, and Exps III and IV attempted to improve estimates for mixture problems. The demonstration simulated points moving at different speeds, tanks filling at different rates, and mixtures of different concentrations. Variations in the simulations resulted in both successful and unsuccessful programs, as measured by how frequently Ss improved their estimates. Variables that influenced the success of a program included successive vs simultaneous presentation of examples, active vs passive viewing, and the similarity between examples and test questions. The quality of feedback received by Ss while learning by viewing, doing, or being coached is compared for these and other programs. (16 ref) (PsycINFO Database Record (c) 2010 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)     

Continued word association in hypothetically psychosis-prone college students.

White male undergraduates who scored deviantly high (2 standard deviations above the mean) on the Physical Anhedonia Scale, the Perceptual Aberration/Magical Ideation (Per/Mag) Scale, or the Nonconformity Scale were compared with controls on either a structured (n?=?63) or an unstructured (n?=?81) continued word-association task. This task has often been used as a measure of psychotic thought disorder. On the unstructured word-association task, Per/Mag Ss produced proportionately more unusual idiosyncratic responses, proportionately fewer common responses, fewer popular responses, and lower response commonality scores than did controls, and these differences were due to those Per/Mag Ss who had also scored at least 1 standard deviation above the mean on the Nonconformity Scale. These findings show mild cognitive slippage in these Ss. Results support the validity of the Per/Mag Scale as a measure of psychosis proneness and the validity of the Nonconformity Scale as a potentiator in the identification of psychosis proneness. (48 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Teaching children to use schematic drawings to solve addition and subtraction word problems.

Two classes of second graders of average and above-average mathematics ability were taught to use differing schematic drawings to represent differing categories of addition and subtraction word problems. Children entered the three-digit numbers used in the problems into the schematic drawings and then were to use the drawings to facilitate the choice of the solution procedure. The children were able to make the correct drawing for a given category, usually inserted the numbers from the problem into a schematic drawing correctly, and usually selected the correct solution strategy for the problem. There was little support for the hypotheses that children use a single part-part-whole schema to solve either all categories of problems or the more difficult "Change" problems. The most difficult problems were those in which the underlying semantic subtractive problem category ("Change-Get-Less" and "Compare") conflicted with the addition solution strategy required to solve the problem. The good-to-excellent posttest performance on most of the possible kinds of addition and subtraction word problems indicates that most of these problems are within the zone of proximal development of second graders of average and above-average mathematics ability. Thus American textbooks can include many of the more difficult word problems, as do textbooks in the Soviet Union. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Training clinical students in personality theory.

In training graduate clinical psychology students at the University of North Dakota, "Using Hall and Lindzey's Theories of Personality as a basic text, the student studies each of the major theorists and is required to write a confidential personality evaluation of himself within the framework of the theory under consideration… . Aside from making the course more meaningful personally, students are found to become more introspective and to raise questions about their role in the clinical situation without ever having been exposed to such notions as counter-transference in any formal sense. The positive transfer to the course in projective techniques is also noteworthy." The student seems to approach clinical report writing in a more mature manner; he recognizes the advantages and inadequacies of a variety of personality theories. "To us it seems that a course in personality theory is one of the most fundamental in the training of clinical psychologists and can be enriched by relating the formulations of various theorists to the personal life of the student." (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

On the word preferences of suicidal versus nonsuicidal college students.

Used a word preference format to investigate reactions to verbal stimuli of 21 suicidal and 21 nonsuicidal university students matched for age and sex. Six words with either aggressive or submissive denotative meanings significantly differentiated the 2 groups. In addition, the word suicide was selected at a higher frequency level by suicidal individuals when compared to their nonsuicidal counterparts. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Children's solution processes in elementary arithmetic problems: Analysis and improvement.

Using error analysis and individual interviews, the problem-solving actions of 176 1st and 2nd graders were analyzed in Exp I. Shortcomings of Ss' knowledge and solution strategies were discovered. It seemed that these shortcomings could be overcome by instruction; therefore, a teaching experiment (Exp II; 52 2nd graders) was undertaken wherein instruction was given for 2 wks to an experimental class, while in a control group, the usual arithmetic program was taught. Experimental instruction related mainly to 3 topics: the equality sign, the part–whole relation, and verification of the outcome of an arithmetic operation. Results show that the experimental teaching program led to a decrease in Ss' thinking errors on elementary addition and subtraction problems. (35 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Training of "third world" students to function as counselors.

Provided more responsive counseling services to ethnic minority students by selecting 20 minority undergraduates (Chicanos, Asian-Americans, blacks, and American Indians) to function as counselors for other minority individuals. Ss were chosen from among 70 who successfully completed a pilot minority-counseling project course at the University of California, Los Angeles. Despite initial difficulties in developing trust and in defining the goals of the program, it is concluded that the training and use of minority-group paraprofessionals are feasible alternatives to current mental health services on campuses. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Gender differences in solution of algebraic word problems containing irrelevant information.

In the 1st experiment, 219 boys and 217 girls in 10th grade classified algebraic word problems in terms of whether the problems contained missing, sufficient, or irrelevant information for solution. Among students with similar levels of general mathematical ability, girls were less likely than boys to identify missing or irrelevant information within problems. More girls than boys perceived irrelevant information within the text of a problem as being necessary for solution. In the 2nd experiment, 11th-grade girls (n?=?234), who were as able as boys (n?=?287) to solve algebraic word problems containing sufficient information, had lower solution rates than did boys on problems containing irrelevant information. On the latter problems, the girls more often incorporated the irrelevant information into their attempted solution than did the boys. The results point to differences between boys and girls in knowledge of problem structure. (PsycINFO Database Record (c) 2010 APA, all rights reserved)