Understanding and using principles of arithmetic: operations involving negative numbers

Previous work has investigated adults' knowledge of principles for arithmetic with positive numbers ( Dixon, Deets, & Bangert, 2001 ). The current study extends this past work to address adults' knowledge of principles of arithmetic with a negative number, and also investigates links between knowledge of principles and problem representation. Participants ( N = 44) completed two tasks. In the Evaluation task, participants rated how well sets of equations were solved. Some sets violated principles of arithmetic and others did not. Participants rated non-violation sets higher than violation sets for two different principles for subtraction with a negative number. In the Word Problem task, participants read word problems and set up equations that could be used to solve them. Participants who displayed greater knowledge of principles of arithmetic with a negative number were more likely to set up equations that involved negative numbers. Thus, participants' knowledge of arithmetic principles was related to their problem representations.     

Leave Her out of It: Person‐Presentation of Strategies is Harmful for Transfer

A common practice in textbooks is to introduce concepts or strategies in association with specific people. This practice aligns with research suggesting that using “real‐world” contexts in textbooks increases students’ motivation and engagement. However, other research suggests this practice may interfere with transfer by distracting students or leading them to tie new knowledge too closely to the original learning context. The current study investigates the effects on learning and transfer of connecting mathematics strategies to specific people. A total of 180 college students were presented with an example of a problem‐solving strategy that was either linked with a specific person (e.g., “Juan's strategy”) or presented without a person. Students who saw the example without a person were more likely to correctly transfer the novel strategy to new problems than students who saw the example presented with a person. These findings are the first evidence that using people to present new strategies is harmful for learning and transfer.     

Conceptual Knowledge,Procedural Knowledge,and Metacognition in Routine and Nonroutine Problem Solving

When, how, and why students use conceptual knowledge during math problem solving is not well understood. We propose that when solving routine problems, students are more likely to recruit conceptual knowledge if their procedural knowledge is weak than if it is strong, and that in this context, metacognitive processes, specifically feelings of doubt, mediate interactions between procedural and conceptual knowledge. To test these hypotheses, in two studies (Ns = 64 and 138), university students solved fraction and decimal arithmetic problems while thinking aloud; verbal protocols and written work were coded for overt uses of conceptual knowledge and displays of doubt. Consistent with the hypotheses, use of conceptual knowledge during calculation was not significantly positively associated with accuracy, but was positively associated with displays of doubt, which were negatively associated with accuracy. In Study 1, participants also explained solutions to rational arithmetic problems; using conceptual knowledge in this context was positively correlated with calculation accuracy, but only among participants who did not use conceptual knowledge during calculation, suggesting that the correlation did not reflect “online” effects of using conceptual knowledge. In Study 2, participants also completed a nonroutine problem-solving task; displays of doubt on this task were positively associated with accuracy, suggesting that metacognitive processes play different roles when solving routine and nonroutine problems. We discuss implications of the results regarding interactions between procedural knowledge, conceptual knowledge, and metacognitive processes in math problem solving.     

Children's understanding of the arithmetic concepts of inversion and associativity

Previous studies have shown that even preschoolers can solve inversion problems of the form a+b-b by using the knowledge that addition and subtraction are inverse operations. In this study, a new type of inversion problem of the form d x e/e was also examined. Grade 6 and 8 students solved inversion problems of both types as well as standard problems of the form a+b-c and d x e/f. Students in both grades used the inversion concept on both types of inversion problems, although older students used inversion more frequently and inversion was used most frequently on the addition/subtraction problems. No transfer effects were found from one type of inversion problem to the other. Students who used the concept of associativity on the addition/subtraction standard problems (e.g., a+b-c=[b-c]+a) were more likely to use the concept of inversion on the inversion problems, although overall implementation of the associativity concept was infrequent. The findings suggest that further study of inversion and associativity is important for understanding conceptual development in arithmetic.     

Children's understanding of the inverse relation between multiplication and division

Children's understanding of the inversion concept in multiplication and division problems (i.e., that on problems of the form d * e/e no calculations are required) was investigated. Children in Grades 6, 7, and 8 completed an inversion problem-solving task, an assessment of procedures task, and a factual knowledge task of simple multiplication and division. Application of the inversion concept in the problem-solving task was low and constant across grades. Most participants approved of the inversion-based shortcut but only a slight majority preferred it. Three clusters of children were identified based on their performance on the three tasks. The inversion cluster used and approved of the inversion shortcut the most and had high factual knowledge. The negation cluster used the negation strategy, had lower approval of the inversion shortcut, and had medium factual knowledge. The computation cluster used computation and had the lowest approval and the weakest factual knowledge. The findings highlight the importance of addressing the multiplication and division inversion concept in theories of children's mathematical competence.     

Core knowledge and its limits: The domain of food

Adults, preschool children, and nonhuman primates detect and categorize food objects according to substance information, conveyed primarily by color and texture. In contrast, they perceive and categorize artifacts primarily by shape and rigidity. The present experiments investigated the origins of this distinction. Using a looking time procedure, Experiment 1 extended previous findings that rhesus macaques (Macaca mulatta) generalize learning about novel food objects by color over changes in shape. Six additional experiments then investigated whether human infants show the same signature patterns of perception and generalization. Nine-month-old infants failed to detect food objects in accord with their intrinsic properties, in contrast to rhesus monkeys tested in previous research with identical displays. Eight-month-old infants did not privilege substance information over other features when categorizing foods, even though they detected and remembered this information. Moreover, infants showed the same property generalization patterns when presented with foods and tools. The category-specific patterns of perception and categorization shown by human adults, children, and adult monkeys therefore were not found in human infants, providing evidence for limits to infants’ domains of knowledge.     

You’ll see what you mean: Students encode equations based on their knowledge of arithmetic

This study investigated the roles of problem structure and strategy use in problem encoding. Fourth-grade students solved and explained a set of typical addition problems (e.g., ) and mathematical equivalence problems (e.g., or ). Next, they completed an encoding task in which they reconstructed addition and equivalence problems after viewing each for 5 s. Equivalence problems of the form overlap with a perceptual pattern found in traditional arithmetic problems (i.e., answer blank in final position), and students’ encoding was poorest on problems of this type. Individual differences in encoding the equivalence problems were related to variations in strategy use. Some students solved blank-final equivalence problems using the standard arithmetic strategy of performing all given operations on all given numbers. These students made more errors in encoding problem structure, but fewer errors in encoding the numbers, than did students who solved the problems using correct or other incorrect strategies. Moreover, students who expressed many strategies for solving the blank-final equivalence problems made fewer errors in encoding problem structure, but more errors in encoding the numbers, than did students who expressed only a single strategy. Results highlight that encoding is intended to guide action and that prior experience can simultaneously facilitate and interfere with accurate encoding.     

To feel is to know relations: James' concept of stream of thought and contemporary studies on procedural knowledge

The theory of William James concerning the temporal and dynamic nature of mind is analyzed as implying that thought is a flow of subjective experience that belongs to the material flow of living beings, and therefore, that knowledge is primarily affective and practical rather than declarative and contemplative. In this context, we will discuss contemporary theory and research relevant to the discussion about declarative and procedural knowledge, with the focus on a literature review in the neurosciences of knowledge. Then we reconstruct James' theory of mind as flow, in terms of relatedness, feeling, and temporality of experience. The Principles suggest that declarative knowledge is not independent, but derived and supported by a more basic knowledge that is both procedural and affective in nature. Finally, we discuss possible lesson for nowadays efforts to develop a dynamic account of the procedural nature of knowledge.     

Children’s understanding of addition and subtraction concepts

After the onset of formal schooling, little is known about the development of children’s understanding of the arithmetic concepts of inversion and associativity. On problems of the form a + b − b (e.g., 3 + 26 − 26), if children understand the inversion concept (i.e., that addition and subtraction are inverse operations), then no calculations are needed to solve the problem. On problems of the form a + b − c (e.g., 3 + 27 − 23), if children understand the associativity concept (i.e., that the addition and subtraction can be solved in any order), then the second part of the problem can be solved first. Children in Grades 2, 3, and 4 solved both types of problems and then were given a demonstration of how to apply both concepts. Approval of each concept and preference of a conceptual approach versus an algorithmic approach were measured. Few grade differences were found on either task. Conceptual understanding was greater for inversion than for associativity on both tasks. Clusters of participants in all grades showed that some had strong understanding of both concepts, some had strong understanding of the inversion concept only, and others had weak understanding of both concepts. The findings highlight the lack of developmental increases and the large individual differences in conceptual understanding on two arithmetic concepts during the early school years.     

The different representational frameworks underpinning abstract and concrete knowledge: Evidence from odd-one-out judgements

Recent evidence from neuropsychological investigations of individuals with global aphasia and deep or deep-phonological dyslexia suggests that abstract and concrete concepts are underpinned by qualitatively different representational frameworks. Abstract words are represented primarily by their association to other words, whilst concrete words are represented primarily by their taxonomic similarity to one another. In the current study, we present the first evidence for this association/similarity distinction to be gathered from healthy research participants. Using a semantic odd-one-out task, it is shown that normal participants identify associative connections more quickly than similarity-based connections when processing abstract words, but that the pattern is reversed for concrete words. It is also demonstrated that the typical concrete-word advantage observed in many cognitive tasks is abolished and even reversed when participants have to comprehend the semantic associations between words. The data provide converging evidence for the different representational frameworks hypothesis and suggest that claims based on information from previous neuropsychological investigations can be generalized to normal cognition.     

Perfectionism: The good,the bad,and the creative

The influence of adaptive and maladaptive perfectionism on creativity was examined. Initially, six measures of creativity were administered, including creative self-perceptions, behavior, and performance measures. Adaptive perfectionism was weakly positively related to creativity, whereas maladaptive perfectionism was unrelated to creativity across five of the six measures. A follow-up study assessed whether initial findings could be generalized to an everyday problem-solving task. Results indicated that adaptive perfectionism was related to higher quality but not originality of solutions. Further, a curvilinear relationship in the shape of an inverted “U” occurred between adaptive perfectionism and four of eight creativity measures. Overall, adaptive perfectionism was consistently, albeit weakly, related to creativity across various types of measures, whereas maladaptive perfectionism was not related to creativity.     

The plausibility of spiritual intelligence: spiritual experience,problem solving and neural sites

Australian teachers in Church related schools have begun to use the term ‘spiritual intelligence’ in their educational discourse. Is it accurate to describe spirituality as a form of intelligence? This paper explores whether the notion of spiritual intelligence is plausible. It addresses this firstly by discussing the notion of spiritual experience as a mechanism for problem solving—one of the central themes that underlies the concept of intelligence. Secondly, it examines some of the neural sites of the human brain that have been found to be active in those who apperceive spiritual experience. In light of this discussion, this paper argues that although some concerns prevail in considering spirituality as a form of intelligence, the concept of spiritual intelligence may nonetheless be rendered as plausible.     

Self-construal level and voice procedures: The individual self as psychological basis for procedural fairness effects

In the current article, we investigate the influence of self-construal level on procedural fairness effects, that is, the finding that fair versus unfair procedures influence people’s evaluations of their relation with decision-making authorities. In two experiments, we manipulated self-construal level by activating the individual self (“I”) or the social self (“We”), and we induced a control condition. Furthermore, we manipulated procedural fairness by granting versus denying participants an opportunity to voice their opinion in a decision-making process. Results consistently revealed stronger procedural fairness effects if the individual self is activated than if the social self is activated. It is concluded that sometimes the individual self, rather than the social self, constitutes the psychological basis for procedural fairness effects.     

Mistakes on display: Incorrect examples refine equation solving and algebraic feature knowledge

Although findings from cognitive science have suggested learning benefits of confronting errors (Metcalfe, 2017), they are not often capitalized on in many mathematics classrooms (Tulis, 2013). The current study assessed the effects of examples focused on either common mathematical misconceptions and errors or correct concepts and procedures on algebraic feature knowledge and solving quadratic equations. Middle school algebra students (N = 206) were randomly assigned to four conditions. Two errorful conditions either displayed errors and asked students to explain or displayed correct solutions and primed students to reflect on potential errors by problem type. A correct example condition and problem-solving control group were also included. Studying and explaining common errors displayed in incorrect examples improved equation-solving ability. An aptitude-by-treatment interaction revealed that learners with limited understandings of algebraic features demonstrated greater benefits. Theoretical implications about using examples to promote learning from errors are considered in addition to suggestions for educational practice.     

The egocentric nature of procedural justice: Social value orientation as moderator of reactions to decision-making procedures

In four studies, the authors investigated the individual-oriented versus social-oriented nature of procedural justice effects by comparing fairness-based responses to decision-making procedures among proself versus prosocial oriented individuals. In Studies 1 through 3, we measured participants’ social value orientation and manipulated whether or not they were granted or denied voice in a decision-making process. Results consistently revealed that the effects of voice versus no-voice on fairness-based perceptions, emotions, and behavioral intentions were significantly more pronounced for individuals with proself orientations than for individuals with prosocial orientations. These findings were extended in Study 4, a field study in which perceived procedural justice was a stronger predictor of satisfaction and organizational citizenship behaviors among proselfs than among prosocials. These findings suggest that procedural justice effects can be accounted for by self-oriented motives or needs, rather than prosocial motives that are often conceptualized as being associated with justice.     

Deconstructing the tower: Parameters and predictors of problem difficulty on the Tower of London task

The Tower of London (TOL) task has been widely used in both clinical and research realms. In the current study, 104 healthy participants attempted all possible moderate- to high-difficulty TOL problems in order to determine: (1) optimal measures of problem solving performance, (2) problem characteristics, other than the minimum moves necessary to solve the problem, that determine participants’ difficulty in solving problems successfully, quickly, and efficiently, and (3) effects of increased task experience on which problem characteristics determine problem difficulty. A factor analysis of six performance measures found that, regardless of task experience, problem difficulty could be captured well either by a single factor corresponding to general quality of solution or possibly by three subordinate factors corresponding to solution efficiency, solution speed, and initial planning speed. Regression analyses predicting these performance factors revealed that in addition to a problem’s minimum moves three problem parameters were critical in determining the problem difficulty: goal position hierarchy, start position hierarchy, and number of solution paths available. The relative contributions of each of the characteristics strongly depended on which performance factor defined performance. We conclude that TOL problem performance is multifaceted, and that classifying problem difficulty using only the minimum moves necessary to solve the problem is inadequate.     

Young children’s acquisition of knowledge about the Earth: A longitudinal study

This longitudinal study examined the acquisition of early knowledge of astronomy to determine whether children’s knowledge at any point in time is consistent with a naive “mental model.” Children were first assessed by means of open questions and drawing tasks at 2 and 3 years of age (N = 143). The knowledge was reassessed over the course of the following 3 years. The results showed that although a few indications of naive mental models were found, in most cases young children’s knowledge was fragmented and accurate knowledge was often expressed alongside inaccurate/synthetic ideas. Furthermore, it was shown that children need to know scientific facts before they start taking the global perspective when describing the world and, when faced with ambiguous open questions, children often experience difficulties that can induce them to change the types of answers they provide.     

Being perceived as a knowledge sender or knowledge receiver: A multistudy investigation of the effect of age on knowledge transfer

As a result of demographic changes, workforces are becoming older and more age diverse. While interactions between workers from different age groups can provide opportunities for mutual learning through bidirectional knowledge transfer, research has yet to investigate how age influences knowledge transfer between age‐diverse colleagues. Building on the organizational theory of age effects, we conducted two studies to examine how age influenced the roles assigned to individuals in knowledge transfer processes, that is, whether they were perceived as knowledge senders or knowledge recipients. In Study 1, we used an experimental vignette design with 450 employees to assess how age affected perceived ability and motivation to share and receive knowledge. Further, we tested the extent to which trustworthiness moderated these relationships. In Study 2, we extended these findings using a dyadic research design with data from 53 age‐diverse knowledge transfer dyads. We examined through which mechanisms the age of one's colleague affected one's knowledge transfer behaviour. We found that the age of one's colleague had a positive effect on one's knowledge receiving behaviour and a negative effect on one's knowledge sharing behaviour. Further, perceived ability to receive knowledge and perceived motivation to share knowledge mediated these effects.     

Autobiographical memory specificity and the persistence of depressive symptoms in HIV-positive patients: Rumination and social problem-solving skills as mediators

Individuals infected with human immunodeficiency virus (HIV) are at elevated risk for depressive conditions, which in turn can negatively impact health-related behaviours and the course of illness. The present study tested the role of autobiographical memory specificity and its interaction with perceived stress in the persistence of depressive symptoms among dysphoric HIV-positive individuals. Additionally, we examined whether rumination and social problem solving mediated these effects. Results indicated that memory specificity moderated the impact of perceived stress, such that perceived stress was more strongly associated with follow-up depressive symptoms among those with greater memory specificity. Rumination, but not social problem solving, mediated this effect. Implications of these findings are discussed.     

Creative Thinking: Processes,Strategies, and Knowledge

Creative achievements are the basis for progress in our world. Although creative achievement is influenced by many variables, the basis for creativity is held to lie in the generation of high‐quality, original, and elegant solutions to complex, novel, ill‐defined problems. In the present effort, we examine the cognitive capacities that make creative problem‐solving possible. We argue that creative problem‐solving depends on the effective execution of a set of complex cognitive processes. Effective execution of these processes is, in turn, held to depend on the strategies employed in process execution and the knowledge being used in problem‐solving. The implications of these observations for improving creative thinking are discussed.