Effects of principle explanation and superficial similarity on analogical mapping in problem solving.

When solving a problem, people often make use of an earlier problem by mapping the objects from the earlier problem to the current one. Three experiments varied the superficial similarity between study and test problems to discriminate 2 views of the mapping process: direct mapping and near miss. Subjects studied 4 probability principles and study problems and solved test problems. The mapping of earlier problems on the basis of superficial similarity would lead to incorrect answers. In Experiment 1, evidence was found for the direct mapping view: Test problems with more similar objects to the study problems were more likely to be (inappropriately) mapped. However, in Experiment 2, in which the principle explanation was embedded in the study problem, this effect was reversed. In Experiment 3, 2 explanations for the differences in effect were contrasted. The discussion focuses on how principle explanation may affect analogical problem solving. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Fact-retrieval automatization and math problem solving by learning disabled, attention-disordered, and normal adolescents.

Learning disabled, attention-disordered, and normal adolescents were assessed with 2 performance measures and 3 behavioral measures, recorded while the Ss worked on math problems. Differences were found between the combined experimental and comparison groups in retrieval speed for each operation, but not in accuracy. Off-task attention contributed to the slower speeds of Ss only during multiplication. Rapid fact retrieval was the only significant predictor of the number of correct answers to word problems. Because of the statistical and procedural controls used (IQ, reading skill, and initial math ability), experimental vs control group differences in problem solving were demonstrated only for specific types of word problems. Overall, these findings support theoretical predictions that attention-disordered and learning disabled children have difficulty with repetitive stimuli, which contributes to basic- and advanced-level math deficits. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

What responses to domain-specific analogy problems reveal about emerging competence: A new perspective on an old acquaintance.

The authors conducted 2 studies to investigate the categorization of students' correct and incorrect responses to domain-specific analogy problems. In Study 1, participants were 6th graders who generated answers to classical analogy problems in the domain of human biology (e.g., cabin:log::skeleton:____). Study 2 focused on the analogical problem solving of undergraduates working in the domain of educational psychology. The basis for categorizing responses in both studies was a 7-level hierarchical response model proposed by P. A. Alexander (1990). Cross-study results provided support for the response categories and for their hierarchical nature. Study 1 indicated that students" answers to analogies were nonrandom in that there were within-subject error preferences. In Study 2, student responses reflected training in domain concepts and strategic processes, and the direction of changes matched expectations of the ordered hierarchical response scheme. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Do self-fulfilling prophecies accumulate, dissipate, or remain stable over time?

The authors examined whether self-fulfilling prophecies accumulate, dissipate, or remain stable over time by using data from more than 500 6th- through 12th-grade students in public school math classes. The authors used multiple regression analyses to assess the extent to which teacher perceptions predicted students' final math marks and standardized math-test scores from 6th through 12th grade. Control variables included 5 measures of student motivation and 2 measures of previous achievement. The results were consistent with both the dissipation and stability hypotheses. Implications for understanding the extent to which social perception creates social reality and the role of expectations in social problems are discussed. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

The role of conceptual understanding in children's addition problem solving

The study examined the relationship between children's conceptual understanding and addition problem-solving procedures. Forty-eight 6- to 8-year-olds solved addition problems and, in a 2nd task, were prompted to judge whether a puppet could use the arithmetic properties of one problem to solve the next problem. Relational properties between consecutive problems were manipulated to reflect aspects of additive composition, commutativity, and associativity principles. Conceptual understanding was assessed by the ability to spontaneously use such relational properties in problem solving (Task 1) and to recognize and explain them when prompted (Task 2). Results revealed that conceptual understanding was related to using order-indifferent, decomposition, and retrieval strategies and speed and accuracy in solving unrelated problems. The importance of conceptual understanding for addition development is discussed.     

From poor performance to success under stress: Working memory, strategy selection, and mathematical problem solving under pressure.

Two experiments demonstrate how individual differences in working memory (WM) impact the strategies used to solve complex math problems and how consequential testing situations alter strategy use. In Experiment 1, individuals performed multistep math problems under low- or high-pressure conditions and reported their problem-solving strategies. Under low-pressure conditions, the higher individuals' WM, the more likely they were to use computationally demanding algorithms (vs. simpler shortcuts) to solve the problems, and the more accurate their math performance. Under high-pressure conditions, higher WM individuals used simpler (and less efficacious) problem-solving strategies, and their performance accuracy suffered. Experiment 2 turned the tables by using a math task for which a simpler strategy was optimal (produced accurate performance in few problem steps). Now, under low-pressure conditions, the lower individuals' WM, the better their performance (the more likely they relied on a simple, but accurate, problem strategy). And, under pressure, higher WM individuals performed optimally by using the simpler strategies lower WM individuals employed. WM availability influences how individuals approach math problems, with the nature of the task performed and the performance environment dictating skill success or failure. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Where should "you" go in a math compare problem?

This study examined the cognitive effects of self-referencing in math word problems in 100 3rd, 4th, and 5th graders. Two types of compare problems were used: compare unknown (CU) and referent unknown (RU). The word you was placed in the problems either as the known or the unknown term. For the CU problems, self-referencing facilitated students' performance regardless of the position of the you term. When self-referencing was applied, students asked for fewer repeats and solved these CU problems faster and with greater accuracy. For the RU problems, however, students benefited from self-referencing only when the self term was placed as the compare (known) term. When the you term was placed as the referent term, the facilitative effect of self-referencing disappeared. The position of the you term in a RU problem apparently has an impact on the translation procedure required in solving the problem. Further research on the cognitive processing issues raised by these data is suggested and the educational implications of the findings are discussed. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Development of the isolation of variables scheme in experimental and "natural experiment" contexts.

The ability of 80 4th, 5th, and 6th graders and college students to logically include and exclude variables in a multivariable natural experiment situation was assessed. It was argued that this situation has somewhat greater external validity than do the traditional Inhelder and Piaget problems used to assess ability to isolate variables. Results indicate that individuals only gradually develop the ability to isolate variables in the natural experiment situation, and, prior to its development, show the same types of errors of reasoning in terms of multiple variables as have been observed in the Inhelder and Piaget experimental problems. Competence in the 2 types of problems emerges during the same age range, though development of competence in the one type of problem and development of competence in the other were found not to be closely related, suggesting that the 2 types of situations pose somewhat different obstacles for Ss. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Finding the Critical Facts: Children's Visual Scan Patterns When Solving Story Problems That Contain Irrelevant Information.

Two experiments were conducted to understand the processes through which 5th graders discriminate relevant from irrelevant information when solving mathematical story problems. Visual scanning was recorded and coded as directed toward relevant information, irrelevant information, the question, workspace, and elsewhere. Experiment 1 focused on the effects of irrelevant numerical information, and Experiment 2 focused on irrelevant qualitative information. The visual scanning results showed that higher achieving and lower achieving students generally used question-guided comparisons of the relevant and irrelevant information when they succeeded. A 2nd effective strategy was to discriminate during their initial reading of the problems. In addition, whereas higher achieving students flexibly varied their visual scanning to fit with problem difficulty, the lower achievers showed less flexibility. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Individual differences in strategy use on division problems: Mental versus written computation.

Individual differences in strategy use (choice and accuracy) were analyzed. A sample of 362 Grade 6 students solved complex division problems under 2 different conditions. In the choice condition students were allowed to use either a mental or a written strategy. In the subsequent no-choice condition, they were required to use a written strategy. Latent class analysis showed that there were 3 subgroups of students with respect to pattern of strategy choices: primarily using a written strategy (more girls than boys); primarily using a mental strategy (more boys than girls); and using a written strategy on more difficult items but a mental strategy on the easier ones (almost no weak mathematical achievers). Strategy accuracies were analyzed with explanatory item response theory modeling. A between-subjects comparison in the choice condition showed that written strategies were usually more accurate than mental strategies, especially for the weak achievers. A within-subject comparison showed that the performance of students who used mental calculation on a particular item in the choice condition improved if they were required to use a written strategy in the no-choice condition. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

More on the Fragility of Performance: Choking Under Pressure in Mathematical Problem Solving.

In 3 experiments, the authors examined mathematical problem solving performance under pressure. In Experiment 1, pressure harmed performance on only unpracticed problems with heavy working memory demands. In Experiment 2, such high-demand problems were practiced until their answers were directly retrieved from memory. This eliminated choking under pressure. Experiment 3 dissociated practice on particular problems from practice on the solution algorithm by imposing a high-pressure test on problems practiced 1, 2, or 50 times each. Infrequently practiced high-demand problems were still performed poorly under pressure, whereas problems practiced 50 times each were not. These findings support distraction theories of choking in math, which contrasts with considerable evidence for explicit monitoring theories of choking in sensorimotor skills. This contrast suggests a skill taxonomy based on real-time control structures. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Students' help seeking during problem solving: Influences of personal and contextual achievement goals.

The author investigated how different types of achievement goals influence elementary school students' help seeking. Fourth and 5th graders were asked to solve math problems and were given the opportunity to request help from an adult. Goals were conceptualized on 2 nested dimensions: (a) locus of the goal (i.e., personal goals that students held at the beginning of the study vs. contextual goals that characterized the task situation) and (b) emphasis of the goal (i.e., the relative importance of learning vs. performance). Personal learning goals had a positive influence and personal performance goals had a negative influence on the frequency of confirmation requests and on actual problem solving. For students who had strong personal performance goals, a contextual learning goal resulted in more process-related help seeking than did a contextual performance goal. Both types of help seeking (i.e., confirmation and process-related requests) had a positive influence on problem solving. Interactions among goals are discussed. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Strategic consultation: The evolution and application of an efficient approach.

Examines how strategic therapy principles can be applied to consultation practice. These principles focus on a conceptual shift in thinking about a problem and its solution, which allows of more effective and often more innovative solutions to consultation problems. In the 1st principle, failed attempted solutions are seen as reinforcing and maintaining problems. The 2nd principle is the notion that it is necessary to develop solutions that implement 2nd order change. The 3rd concept proposes that it is important to change the consultee's perceptions of the problem in order to assist in developing and implementing new solutions. These principles of strategic consultation can also be applied in many settings and can be incorporated in many other consultation approaches. These principles focus on a conceptual shift in thinking about problems and their solutions. This shift allows for more effective, efficient, and often more innovative solutions to consultation problems. A step-by-step procedure is presented. A case example demonstrates how the approach can be implemented. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

The effects of schema-based instruction on the mathematical word-problem-solving performance of students with learning disabilities

This study examined the effects of a schema-based direct instruction strategy on the word-problem-solving performance of three third- and fourth-grade students (2 girls, 1 boy) with learning disabilities. An adapted multiple-probe-across-students design was used. Results indicated that the intervention was successful in increasing the percentage of correct solutions to word problems for all 3 students. In addition, maintenance of word-problem solving was seen 2 to 3 weeks after the study. Student interviews indicated that the strategy was beneficial. Further research with different students and problem types (e.g., multistep) and an investigation of the long-term effects of the strategy and its use in novel settings appear warranted.     

This is like that: The use of earlier problems and the separation of similarity effects.

Novices attempting to solve a problem often are reminded of an earlier problem that illustrated a principle. Two experiments examined how these earlier problems are used and how this use is related to these remindings. Subjects studied four probability principles with related word problems. Test problems varied in their similarity to the study problems on story lines, objects, and correspondence of objects (variable roles). Experiment 1 tested whether remindings cue the principle or serve as the sources of detailed analogies. When the appropriate formula was provided with each test, the similarity of story lines had no effect, but object correspondences had a large effect. These results support an analogical account in which mapping is affected by the similarity of objects between study and test problems. Experiment 2 began to separate the aspects of similarity affecting the access and use of earlier problems by showing that, with confusable principles, similar story lines increased the access, but did not affect the use. The access appears to be sensitive to the relative similarity of examples because with distinctive principles, similar story lines had little effect. Discussion focuses on the further specification of the processes of noticing and analogical use of earlier problems. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Children's differential performance on deductive and inductive syllogisms.

At what age and in what ways do children distinguish deductive and inductive problems? In Exp 1, students from kindergarten and from Grades 2, 4, and 6 were presented with deductive or inductive inference problems and asked to draw an inference and rate their confidence. By 4th grade, confidence ratings for deductive problems were higher than those for inductive problems, and responses were faster for deductive than for inductive items. In Exp 2, students from Grades 2, 3, 4, and 5 responded to the same problems used in Exp 1 but were asked to provide explanations for their responses. Again, confidence was higher with deductive than with inductive problems, and latency to respond was faster for deductive than for inductive items. Further, explanations differed as a function of the type of problem. These findings help fill in gaps in the emerging picture of the development of children's reasoning skills. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Using technology to explore social networks and mechanisms underlying peer effects in classrooms.

Peer interactions among children have long interested social scientists. Identifying causal peer effects is difficult, and a number of studies have used random assignment to produce evidence that peers affect each other's outcomes. This focus by sociologists and economists on whether peers affect each other has not been matched by direct evidence on how these effects operate. The authors argue that one reason for the small number of studies in sociology and economics on the mechanisms underlying peer effects is the difficulty of collecting data on microinteractions. They argue technology reduces data collection costs relative to direct observation and allows for realistic school activities with randomly assigned peers. The authors describe a novel strategy for collecting data on peer interactions and discuss how this approach might shed light on mechanisms underlying peer influence. The centerpiece of this strategy is the use of handheld computers by middle and high school students as part of interactive math and science lessons called the Discussion Game. The handhelds collect data on interactions between students and track how students' answers evolve as they interact with different peers. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

What do students learn while solving mathematics problems?

Evidence is accumulating that the means–ends problem-solving strategies used conventionally by novice problem solvers are relatively ineffective as vehicles for the acquisition of schemata characteristic of experts. It is suggested that a means–ends strategy places a heavy load on cognitive processing capacity and that this load retards knowledge acquisition. A series of 3 experiments using trigonometry problems and a total of 20 10th-grade and 42 9th-grade students as Ss was carried out. The problem goal was modified with the intention of disrupting the strategy used by novices. It was hypothesized that the development of adequate cognitive representations of the sine, cosine, and tangent ratios would be enhanced as a consequence. Results indicate that preventing novice problem solvers from using means–ends analysis resulted in fewer mathematical errors both during acquisition and on subsequent problems, including transfer problems. This provided some evidence for the contention that a means–ends strategy places a heavy load on cognitive processing capacity, which retards knowledge acquisition. (13 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Mental multiplication skill: Structure, process, and acquisition.

Collected response time (RT) and error data on multiplication problems up to 9?×?9 from 86 Ss in Grades 2–5 and from 60 undergraduate and graduate students. Results show that most errors involved correct products to other problems and that a developmental trend emerged in which the specific errors made by children mirrored adult errors by Grade 5. The error patterns indicate that an associative network evolves in which problem operands become linked to specific sets of candidate answers. Retrieval is governed by a process that activates candidates, and accessibility of correct answers is impeded by competing associations: At all skill levels, both problem-error rates and product-error rates (i.e., how often a problem's correct product occurs as an incorrect response to other problems) contributed to predicting correct problem RT in multiple regression analyses. These interference variables yielded higher correlations than did structural variables (e.g., the numerical size of problem operands), the latter having provided the basis for previous models of arithmetic memory. A network-interference account is proposed that explains the slow course of acquisition and differential problem difficulty in terms of interference by false associations. (French abstract) (37 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Development of intuitive theories of motion: Curvilinear motion in the absence of external forces.

96 4.5–12 yr olds and 20 college students were asked to draw the path a ball would take when exiting a curved tube. As in previous studies, many Ss erroneously predicted curvilinear paths. However, a clear U-shaped curve was evident in the data. Preschoolers and kindergartners performed as well as college students, whereas school-aged Ss were more likely to make erroneous predictions. Results of a 2nd study, with 24 preschoolers (mean age 5 yrs 2 mo), suggest that the youngest children's correct responses could not be attributed to response biases or drawing abilities. This developmental trend suggests that the school-aged children are developing intuitive theories of motion that include erroneous principles. Results are related to the "growth errors" found in other cognitive domains and to the historical development of formal theories of motion. (15 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)