Experience and problem representation in statistics

This research investigated experience level differences in problem representation in statistics. A triad judgment task was designed so that source problems shared either surface similarity (story narrative) or structural (inferential level) features (t test, correlation, or chi-square) with the target problem. Graduate students with varying levels of experience in statistics were asked to choose which source problem "goes best" with the target problem for each triad. Given a choice between a problem that shares surface-level characteristics and one that shares inferential-level characteristics, students who had taken 0 to 4 courses in statistics tended to represent problems on the basis of surface-level features. Students who had more than 4 courses did not consistently make choices on the basis of surface-level features, nor did they consistently rely on structural features. However, all students with statistics course backgrounds noticed structural features when competition between different types of features was eliminated. The role of surface and structural features in determining problem representations is discussed.     

Question Design Affects Students' Sense-Making on Mathematics Word Problems

Mathematics word problems provide students with an opportunity to apply what they are learning in their mathematics classes to the world around them. However, students often neglect their knowledge of the world and provide nonsensical responses (e.g., they may answer that a school needs 12.5 buses for a field trip). This study examined if the question design of word problems affects students' mindset in ways that affect subsequent sense-making. The hypothesis was that rewriting standard word problems to introduce inherent uncertainty about the result would be beneficial to student performance and sense-making because it requires students to reason explicitly about the context described in the problem. Middle school students (N = 229) were randomly assigned to one of three conditions. In the standard textbook condition, students solved a set of six word problems taken from current textbooks. In the modified yes/no condition, students solved the same six problems rewritten so the solution helped answer a “yes” or “no” question. In the disfluency control condition, students solved the standard problems each rewritten in a variety of fonts to make them look unusual. After solving the six problems in their assigned condition, all students solved the same three “problematic” problems designed to assess sense-making. Contrary to predictions, results showed that students in the modified yes/no condition solved the fewest problems correctly in their assigned condition problem set. However, consistent with predictions, they subsequently demonstrated more sense-making on the three problematic problems. Results suggest that standard textbook word problems may be able to be rewritten in ways that mitigate a “senseless” mindset.     

Analogical transfer, problem similarity, and expertise

When we encounter a new problem, we are often reminded of similar problems solved earlier and may use the solution procedure from an old problem to solve a new one. Such analogical transfer, however, has been difficult to demonstrate empirically, even within a single experimental session. This article proposes a framework for conceptualizing analogical problem solving that can account for the conflicting findings in the literature. In addition, the framework leads to two predictions concerning the transfer behavior of experts and novices. These predictions concern both positive and negative transfer and are based on the different types of features included in the problem representations of experts and novices. First, when two problems share structural features but not surface features, spontaneous positive transfer should be more likely in experts than in novices. Second, when two problems share surface but not structural features, spontaneous negative transfer should be stronger for novices than for experts. These predictions were supported by the results of three experiments involving college students solving a complex arithmetic word problem.     

Suppressed semantic information accelerates analytic problem solving

The present study investigated the limits of semantic processing without awareness, during continuous flash suppression (CFS). We used compound remote associate word problems, in which three seemingly unrelated words (e.g., pine, crab, sauce) form a common compound with a single solution word (e.g., apple). During the first 3 s of each trial, the three problem words or three irrelevant words (control condition) were suppressed from awareness, using CFS. The words then became visible, and participants attempted to solve the word problem. Once the participants solved the problem, they indicated whether they had solved it by insight or analytically. Overall, the compound remote associate word problems were solved significantly faster after the problem words, as compared with irrelevant words, were presented during the suppression period. However this facilitation occurred only when people solved with analysis, not with insight. These results demonstrate that semantic processing, but not necessarily semantic integration, may occur without awareness.     

Task‐Related Factors that Influence the Spontaneous Use of Diagrams in Math Word Problems

Diagrams are effective tools for problem solving. However, previous findings indicate that students generally do not use diagrams spontaneously. This study examined task‐related factors that may influence the spontaneity of diagram use. Experiment 1 compared two possible explanations: the first, that the length‐relatedness of the story context of the problem (i.e. whether it involves the measurement of length) determines the likelihood of diagram use; and the second, that the cognitive cost of transforming the situation described in the word problem to an abstract diagrammatic representation is the more important factor. Four math word problems, differing in their story context and structure, were administered to eighth‐grade Japanese students (n = 125) to solve. The results provide support for the cognitive transformation cost explanation. The results of experiment 2, in which the problems were administered to students in both Japan (n = 291) and New Zealand (n = 323), confirm this finding. Copyright © 2011 John Wiley & Sons, Ltd.     

Can “you” make a difference? Investigating whether perspective‐taking improves performance on inconsistent mathematical word problems

Pronouns encouraging a second‐person perspective (e.g., “you/your”) affect peoples' mental representations constructed while reading and improve learning. The present study applied these insights to a domain in that such pronoun effects have yet been unexplored: mathematical word problem solving. Specifically, we encouraged a second‐person perspective (using “your”) in an attempt to reduce the consistency effect, that is, the finding that more errors are made on word problems containing a relational keyword inconsistent rather than consistent with the required arithmetic operation. Primary school children solved consistent and inconsistent word problems (containing the relational keywords “less than”) presented in third‐person (i.e., store name) or second‐person (“your store”) perspective. Results demonstrated the consistency effect, but the perspective manipulation did not produce significant differences between conditions, that is, a second‐person perspective did not reduce the consistency effect. These findings suggest that reducing the consistency effect may require a less subtle approach than using personalized pronouns.     

算术应用题二重变异样例学习的迁移效果

设计并运用了三种表面特征变异的二重样例和三种结构特征变异的二重样例,对二年级小学生解决算术应用题的迁移效果进行了实验研究。结果表明：一个原样例的学习就能明显促进了二年级小学生解决与原样例表面特征不同而结构特征相同的近迁移问题;学习三种表面特征变异的二重样例后没有产生明显的远迁移效果;而学习了三种结构特征变异的二重样例后产生了不同程度的远迁移效果,其中,规则平行组合二重样例学习的远迁移效果最好,其次是规则变异二重样例学习的远迁移效果,规则镶嵌组合变异二重样例学习的远迁移效果最差     

Teaching college students to pour accurately using behavioral skills training: Evaluation of the effects of peer modeling

Excessive alcohol consumption among college students is a serious problem. Alcohol education courses, during which students are taught to free‐pour accurate servings of alcohol, have been proposed as one strategy to reduce this problem. However, few studies have evaluated the effects of peer models on free‐pour skill maintenance. We used a nonconcurrent multiple‐baseline across subjects design to assess the effects of peer modeling on maintenance of college students' (N = 19) pouring skills of a standard serving of beer immediately following Behavioral Skills Training (BST). Results indicated participants who inaccurately poured a standard serving of beer at baseline (n = 17) accurately poured following receipt of BST, and all participants (N = 19) maintained accurate pouring in the presence of peer confederate models who poured inaccurately or accurately. Results suggest BST is an efficient (i.e., low to no‐cost, requires no sophisticated laboratory space or software, and makes use of available and affordable materials) and effective tool for teaching college students to pour standard servings of beer and that the immediate introduction of inaccurately pouring peer models has little or no effect on skill maintenance.     

Boundless Creativity: Evidence for the Domain Generality of Individual Differences in Creativity

This study investigated correlates of domain‐general and domain‐specific components of creativity. 158 college students completed a questionnaire that assessed their motivational and personality traits (i.e., intrinsic and extrinsic motivation, creative personality, and originality in word association) as well as intellectual abilities (SAT verbal and quantitative scores). Under two different instruction conditions (standard instruction or explicit “be creative” instruction), students also took a battery of multi‐item, product‐based tests of creativity in three domains (artistic, verbal, and mathematical creativity). Factor analyses showed evidence of domain‐generality of creativity. Furthermore, results from structural equation models showed that motivational and personality traits and intellectual abilities were associated with the domain‐general component of creativity. Only one variable (SAT quantitative score) was found to be associated with the domain‐specific component of mathematical creativity under the explicit “be creative” instruction condition. These results affirm the domain‐generality of creativity and challenge researchers to identify correlates of domain‐specific components of creativity.     

Interactive effects of numerical surface form and operand parity in cognitive arithmetic

In Experiment 1, adults (n = 48) performed simple addition, multiplication, and parity (i.e., odd-even) comparisons on pairs of Arabic digits or English number words. For addition and comparison, but not multiplication, response time increased with the number of odd operands. For addition, but not comparison, this parity effect was greater for words than for digits. In Experiment 2, adults (n = 50) solved simple addition problems in digit and word format and reported their strategies (i.e., retrieval or procedures). Procedural strategies were used more for odd than even addends and much more for word than digit problems. The results indicate that problem encoding and answer retrieval processes for cognitive arithmetic are interactive rather than strictly additive stages.     

Pupils' over‐reliance on linearity: A scholastic effect?

Background. From upper elementary education on, children develop a tendency to over‐use linearity. Particularly, it is found that many pupils assume that if a figure enlarges k times, the area enlarges k times too. However, most research was conducted with traditional, school‐like word problems. Aims. This study examines whether pupils also over‐use linearity if non‐linear problems are embedded in meaningful, authentic performance tasks instead of traditional, school‐like word problems, and whether this experience influences later behaviour. Sample. Ninety‐three sixth graders from two primary schools in Flanders, Belgium. Method. Pupils received a pre‐test with traditional word problems. Those who made a linear error on the non‐linear area problem were subjected to individual interviews. They received one new non‐linear problem, in the S‐condition (again a traditional, scholastic word problem), D‐condition (the same word problem with a drawing) or P‐condition (a meaningful performance‐based task). Shortly afterwards, pupils received a post‐test, containing again a non‐linear word problem. Results. Most pupils from the S‐condition displayed linear reasoning during the interview. Offering drawings (D‐condition) had a positive effect, but presenting the problem as a performance task (P‐condition) was more beneficial. Linear reasoning was nearly absent in the P‐condition. Remarkably, at the post‐test, most pupils from all three groups again applied linear strategies. Conclusions. Pupils' over‐reliance on linearity seems partly elicited by the school‐like word problem format of test items. Pupils perform much better if non‐linear problems are offered as performance tasks. However, a single experience does not change performances on a comparable word problem test afterwards.     

Categorizing words using ‘frequent frames’: what cross‐linguistic analyses reveal about distributional acquisition strategies

Mintz (2003 ) described a distributional environment called a frame, defined as the co‐occurrence of two context words with one intervening target word. Analyses of English child‐directed speech showed that words that fell within any frequently occurring frame consistently belonged to the same grammatical category (e.g. noun, verb, adjective, etc.). In this paper, we first generalize this result to French, a language in which the function word system allows patterns that are potentially detrimental to a frame‐based analysis procedure. Second, we show that the discontinuity of the chosen environments (i.e. the fact that target words are framed by the context words) is crucial for the mechanism to be efficient. This property might be relevant for any computational approach to grammatical categorization. Finally, we investigate a recursive application of the procedure and observe that the categorization is paradoxically worse when context elements are categories rather than actual lexical items. Item‐specificity is thus also a core computational principle for this type of algorithm. Our analysis, along with results from behavioural studies ( Gómez, 2002 ; Gómez and Maye, 2005 ; Mintz, 2006 ), provides strong support for frames as a basis for the acquisition of grammatical categories by infants. Discontinuity and item‐specificity appear to be crucial features.     

Use of a self-regulated strategy intervention to improve word problem-solving skills of students with mild disabilities

We investigated the effects of self-regulated strategy instruction on the ability of four elementary-school students with mild handicaps (2 with learning disabilities [LD] and 2 with mild mental retardation [MMR]) to perform 4 types of addition and subtraction word problems. We used a multiple baseline design and a self-instructional strategy to address story problem solving. Results indicated that the strategy was effective in increasing the number of correctly solved word problems. Gains were maintained across a 6 and 8 week follow-up check. The study supports previous research indicating that cognitive strategies are effective for students with LD and suggests that cognitive strategies are appropriate for students with MMR.     

小学2～4年级儿童数学应用题表征策略差异的研究

本研究运用实验法和临床访谈法对某普通小学的123名2～4年级学生进行了数学应用题测验,以考察数学学优生和学差生在解决比较应用题时表征策略的差异。结果表明:(1)从2～4年级儿童解答一致和不一致应用题上看,学优生较多地使用问题模型策略对问题进行表征,学差生较多地使用直接转换策略对问题进行表征;(2)除学差女生的解题正确率低于学差男生的正确率,学差女生自我报告中直接转换策略的使用多于学差男生外,在其它方面,性别差异并不显著;(3)随着年级的升高,学优生在使用问题模型策略上越来越成熟,学差生并没有学会使用更加有效的问题模型表征策略,仍然停留在直接转换策略上,但他们在关于策略使用的认识上有所提高。     

Reducing Perceptions of Time Required to Complete Math Assignments by Adding Problems to Assignments: A Synthesis of the Additive Interspersal Research

Results from nine experiments were analyzed to evaluate the relationship between relative problem completion rates (RPCR) and judgments of time involving mathematics assignments. For each experiment, control assignments contained only target mathematics computation or word problems. Matched experimental assignments contained similar target problems with additional, briefer problems interspersed after every third target problem. After working on each assignment for the same amount of time, students selected the assignment that they judged would require the least or most amount of time to complete. Regression analysis revealed a significant linear relationship between RPCR (i.e., PCR on experimental assignments/PCR on control assignments) and time perceptions (r=.97). As RPCR increased, the percentage of students judging the experimental assignments as being less time consuming increased proportionally. Discussion focuses on application of time perception theories to educational settings.     

A Visual Representation to Quantitate,Diagnose, and Improve Creativity in Insight Problem Solving

A new visual representation for insight problems permits 22 new quantitative measures; which leads to a detailed diagnosis of a person's (or team's) creative weaknesses; which then leads to prescribing targeted, effective counter‐techniques for each weakness. Currently, only two measures are consistently used for insight problem solving: the number of problems solved and the time to solve the problems. These coarse measurements do not reveal the intricate dynamics of solving insight problems. Furthermore, four commonly used creativity measures (i.e., fluency, originality, flexibility, and elaboration) are often not applied to insight problems. This new visualization permits the easy application of all four creativity measures. I challenge creativity researchers to help determine which of the 22 proposed quantitative measures are the most diagnostic for insight problem solving in isolation and, in a weighted linear combination, which might yield an effective quotient (i.e., overall measure) of insight problem solving ability.     

Calculation latency: The μ of memory and the τ of transformation

Does numeral format (e.g., 4 + 8 vs. four + eight) affect calculation per se? University students (N=47) solved single-digit addition problems presented as Arabic digits or English words and reported their strategies (memory retrieval or procedures such as counting or transformation). Decomposition of the response time (RT) distributions into μ (reflecting shift) and t (reflecting skew) confirmed that retrieval trials contributed predominantly to μ, whereas procedure trials contributed predominantly to τ. The format × problem size RT interaction (i.e., greater word-format RT costs for large problems than for small problems) was associated entirely with μ and not with τ. Reported use of procedures presented a corresponding format × size interaction. Together, these results indicate that, relative to the well-practiced digit format, the unfamiliar word format disrupts number-fact retrieval and promotes use of procedural strategies.     

Not Simply “In Their Heads”: Individual Differences Associated With Victimization and Health1

Research has not examined whether victimization predicts health after controlling for personality differences associated with victimization and health. In Study 1, college students (N = 1182) completed surveys assessing victimization, health, and personality. In Study 2, college students (N = 69) participated in a short‐term longitudinal study that examined whether (a) victimization would predict health changes; (b) increases in victimization would lead to increased health problems; and (c) fall health would predict spring victimization when personality was considered. After controlling for individual differences, we found victimization predicted health outcomes (Study 1) and health problems over time (Study 2). These findings suggest the victimization–health link may occur because of physiological changes, rather than personality differences associated with victimization and health.     

Familiar words can serve as a semantic seed for syntactic bootstrapping

Young children can exploit the syntactic context of a novel word to narrow down its probable meaning. But how do they learn which contexts are linked to which semantic features in the first place? We investigate if 3‐ to 4‐year‐old children (n = 60) can learn about a syntactic context from tracking its use with only a few familiar words. After watching a 5‐min training video in which a novel function word (i.e., ‘ko’) replaced either personal pronouns or articles, children were able to infer semantic properties for novel words co‐occurring with the newly learned function word (i.e., objects vs. actions). These findings implicate a mechanism by which a distributional analysis, associated with a small vocabulary of known words, could be sufficient to identify some properties associated with specific syntactic contexts.     

Relationships Between Treatment Attitudes,Psychological Symptoms,Emotional Competence,and Help‐Seeking Intentions

The authors examined whether emotional competence (i.e., awareness, coping strategies) predicted help‐seeking intentions above and beyond previously identified factors (i.e., attitudes and perceived stigma toward professional help, psychological symptom severity) in an undergraduate sample (N = 531). Emotional awareness predicted help‐seeking intentions for personal/emotional problems and suicidal thoughts. Emotional coping strategies predicted help‐seeking intentions for suicidal thoughts and moderated the relationship between symptom severity and help‐seeking intentions for suicidal thoughts. Efforts to increase help‐seeking should address students' emotional competence.