Individual differences in cognitive arithmetic

Unities in the processes involved in solving arithmetic problems of varying operations have been suggested by studies that have used both factor-analytic and information-processing methods. We designed the present study to investigate the convergence of mental processes assessed by paper-and-pencil measures defining the Numerical Facility factor and component processes for cognitive arithmetic identified by using chronometric techniques. A sample of 100 undergraduate students responded to 320 arithmetic problems in a true-false reaction-time (RT) verification paradigm and were administered a battery of ability measures spanning Numerical Facility, Perceptual Speed, and Spatial Relations factors. The 320 cognitive arithmetic problems comprised 80 problems of each of four types: simple addition, complex addition, simple multiplication, and complex multiplication. The information-processing results indicated that regression models that included a structural variable consistent with memory network retrieval of arithmetic facts were the best predictors of RT to each of the four types of arithmetic problems. The results also verified the effects of other elementary processes that are involved in the mental solving of arithmetic problems, including encoding of single digits and carrying to the next column for complex problems. The relation between process components and ability measures was examined by means of structural equation modeling. The final structural model revealed a strong direct relation between a factor subsuming efficiency of retrieval of arithmetic facts and of executing the carry operation and the traditional Numerical Facility factor. Furthermore, a moderate direct relation between a factor subsuming speed of encoding digits and decision and response times and the traditional Perceptual Speed factor was also found. No relation between structural variables representing cognitive arithmetic component processes and ability measures spanning the Spatial Relations factor was found. Results of the structural modeling support the conclusion that information retrieval from a network of arithmetic facts and execution of the carry operation are elementary component processes involved uniquely in the mental solving of arithmetic problems. Furthermore, individual differences in the speed of executing these two elementary component processes appear to underlie individual differences on ability measures that traditionally span the Numerical Facility factor.(ABSTRACT TRUNCATED AT 400 WORDS)     

Neural correlates of arithmetic calculation strategies

Recent research into math cognition has identified areas of the brain that are involved in number processing (Dehaene, Piazza, Pinel, & Cohen, 2003) and complex problem solving (Anderson, 2007). Much of this research assumes that participants use a single strategy; yet, behavioral research finds that people use a variety of strategies (LeFevre et al., 1996; Siegler, 1987; Siegler & Lemaire, 1997). In the present study, we examined cortical activation as a function of two different calculation strategies for mentally solving multidigit multiplication problems. The school strategy, equivalent to long multiplication, involves working from right to left. The expert strategy, used by “lightning” mental calculators (Staszewski, 1988), proceeds from left to right. The two strategies require essentially the same calculations, but have different working memory demands (the school strategy incurs greater demands). The school strategy produced significantly greater early activity in areas involved in attentional aspects of number processing (posterior superior parietal lobule, PSPL) and mental representation (posterior parietal cortex, PPC), but not in a numerical magnitude area (horizontal intraparietal sulcus, HIPS) or a semantic memory retrieval area (lateral inferior prefrontal cortex, LIPFC). An ACT-R model of the task successfully predicted BOLD responses in PPC and LIPFC, as well as in PSPL and HIPS.     

Young and older adults' strategies in complex arithmetic

Young and older adults solved complex addition problems such as 49 + 56. Verbal protocols, solution times, and percentage errors documented strategy repertoire and strategy selection in young and older participants and age-related differences in complex arithmetic performance. Both young and older adults used a set of 9 strategies to solve complex addition problems, although many older adults used a smaller strategy repertoire. The data also showed age-related differences in strategy execution and strategy selection. We discuss general implications of the present data to further our understanding of complex arithmetic and the role of strategic variations during aging.     

Polysemy and the acquisition of the cognitive internal state lexicon

Previous research (Hall, Scholnick, & Hughes, 1987) suggested that cognitive word meanings denote six increasingly abstract and complex cognitive processes. In this paper, we further investigated two interrelated aspects of this framework. A set of 36 conversations with children aged 4:5 to 5 provided the data. In Study 1, issues related to the optimal ordering of levels were examined by comparing the original sequence with a restructured model. In Study 2, the hypothesis that, in addition toknow, other polysemous cognitive internal state words would be hierarchically organized was tested. The findings lent support to the restructured model, but rejected the hypothesis that polysemous words within the domain were characterized by equivalent,hhierarchically organized levels. We conclude by proposing that cognitive internal state words were related by complementarity of levels rather than equivalence.The research on which this paper is based was supported by a grant from the Carnegie Corporation of New York to William S. Hall. Data analysis was supported by the Computer Science Center at the University of Maryland, College Park.     

Temporal tuning in the acquisition of cognitive skill

Thetemporal tuning hypothesis suggests that individuals adjust the timing of cognitive performances to achieve temporal coordination of mental processes and the data on which they operate, and that this adjustment becomes more precise with practice. Participants in two experiments performed self-paced multiple-step arithmetic tasks in which the information needed for each step was briefly displayed at the participants’ request. Timing constraints were manipulated by varying between subjects the delay between requests and displays of information. In Experiment 1, both operators and operands appeared step by step, and participants achieved a modest degree of temporal adjustment that did not change with practice. In Experiment 2, participants could preview operators while operands appeared step by step. In that experiment, participants achieved more precise temporal adjustment, and the amount of adjustment increased with practice. These results demonstrate the phenomenon of temporal tuning in symbolic cognitive skills and suggest some constraints on the ability to anticipate the time course of one’s mental processes.     

Building blocks for a cognitive science-led epistemology of arithmetic

Philosophical Studies - In recent years philosophers have used results from cognitive science to formulate epistemologies of arithmetic (e.g. Giaquinto in J Philos 98(1):5–18, 2001). Such...     

The acquisition of robust and flexible cognitive skills

The authors introduce a model of skill acquisition that incorporates elements of both traditional models and models based on embedded cognition by striking a balance between top-down and bottom-up control. A knowledge representation is used in which pre- and postconditions are attached to actions. This model captures improved performance due to learning not only in terms of shorter solution times and lower error rates during the task but also in an increased flexibility to solve similar problems and robustness against unexpected events. In 3 experiments using a complex aviation task, the authors contrasted instructions that explicitly stated pre- and postconditions with conventional instructions that did not. The instructions with pre- and postconditions led to better and more robust performance than other instructions, especially on problems that required transfer. The parameters of the model were estimated to obtain a quantitative fit of the results of Experiment 1, which was then successfully used to predict the results of Experiments 2 and 3.     

Rule acquisition events in the discovery of problem-solving strategies

Although there are many machine-learning programs that can acquire new problem-solving strategies, it is not known exactly how their processes will manifest themselves in human behavior, if at all. In order to find out, a line-by-line protocol analysis was conducted of a subject discovering problem-solving strategies. A model was developed that could explain 96% of the lines in the protocol. On this analysis, the subject's learning was confined to 11 rule acquisition events, wherein she temporarily abandoned her normal problem solving and focused on improving her strategic knowledge. Further analysis showed that: (1) Not all rule acquisition events are triggered by impasses; (2) Rules are acquired gradually, both because of competition between new and old rules, and because of the subject's apparently deliberate policy of gradual generalization. (3) This subject took a scientific approach to strategy discovery, even planning and conducting small experiments.     

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.     

Relationship of arithmetic problem solving and reflective-impulsive cognitive styles in third-grade students

Different individuals approach mathematical problems in a variety of ways, with these different approaches also reflected in over-all cognitive styles. This investigation had two purposes, first, to determine whether good and poor arithmetic problem solvers differ substantially in cognitive style, and second, to determine whether the students, after training in techniques of solving arithmetic problems, improve their performance with no significant change in cognitive style. A total of 98 third graders participated (mean age 8.1 yr.; 50 boys, 48 girls). The Matching Familiar Figure Test was used to classify the students by cognitive style as either Reflective or Impulsive. Students also were given training with different problem-solving exercises for different arithmetic problems. The training program in problem-solving strategies did not improve performance on arithmetic problems for Reflective students; however, Impulsive students' performance did improve after training.     

The dynamics of cognition: An ACT-R model of cognitive arithmetic

Zusammenfassung. Forschungsarbeiten zur Kognitiven Arithmetik untersuchen die mentale Repräsentation von Zahlen und arithmetischen Fakten sowie die kognitiven Prozesse die diese generieren, abrufen und manipulieren. Das Spannungsfeld zwischen der scheinbar einfachen formalen Struktur dieses Aufgabenbereichs und den Schwierigkeiten, die Kinder bei seiner Bewältigung haben, stellt einen einzigartigen Zugang zum Studium kognitiver Prozesse dar. Der vorliegende Beitrag präsentiert einen Erklärungsansatz der zentralen Befunde des Forschungsgebietes auf der Grundlage eines ACT-R Modells zur Lebenszeit-Simulation des Erwerbs arithmetischen Wissens. Die Anwendung der Bayesischen Lernmechanismen der ACT-R Architektur zeigen auf, wie sich diese Befunde auf die statistische Struktur des Aufgabengebiets zurückführen lassen. Aus den präzisen Vorhersagen der Simulation werden sowohl Hinweise zur Vermittlung arithmetischen Wissens abgeleitet als auch Erkenntnisse über die Architektur ACT-R selbst gewonnen. Im Rahmen einer formalen Analyse wird gezeigt, daß sich die vorgestellte Simulation als dynamisches System betrachten läßt, dessen Lernergebnis unmittelbar von Parametern der Architektur abhängt. Eine Untersuchung der Sensitivität der Parameter der Simulation belegt, daß die Werte, die zur besten Anpassung an die empirischen Daten führen, auch eine in einer optimalen Performanz resultieren. Die Implikationen dieses Ergebnis für die grundlegende Adaptivität menschlicher Kognition werden diskutiert. Abstract. Cognitive arithmetic studies the mental representation of numbers and arithmetic facts and the processes that create, access, and manipulate them. The contradiction between the apparent straightforwardness of its exact formal structure and the difficulties that every child faces in mastering it provides an important window into human cognition. An ACT-R model is proposed which accounts for the central results of the field through a single simulation of a lifetime of arithmetic learning. The use of the architecture's Bayesian learning mechanisms explains how these effects arise from the statistics of the task. Because of the precise predictions of the simulation, a number of lessons are derived concerning the teaching of arithmetic and the ACT-R architecture itself. A formal analysis establishes that the simulation can be viewed as a dynamical system whose ultimate learning outcome is fundamentally dependent upon some architectural parameters. Finally, an empirical study of the sensitivity of the simulation to its parameters determines that the values that yield the best fit to the data also provide optimal performance. The implications of these findings for the fundamental adaptivity of human cognition are discussed.     

The cognitive mediation of thought-control strategies

Research suggests that people use various strategies to control their normally occurring intrusive thoughts. Strategies that involve worrying about the thought and self punishment appear to be associated with certain forms of psychopathology, such as obsessive-compulsive (OC) symptoms. The present study sought to examine whether dysfunctional beliefs associated with OC symptoms (e.g., beliefs that intrusive thoughts are highly significant) underlie the use of such thought control strategies. Ninety-three non-clinical participants completed self-report questionnaires measuring cognitive variables, thought-control strategies, and OC symptoms. Analyses revealed that overestimates of threat and responsibility, beliefs about the significance and need to control intrusions, the need for perfection and certainty, and scrupulosity were associated with the use of punishment, but not worry thought control strategies. These cognitive phenomena also mediated the relationship between OC symptoms and the use of punishment as a thought-control strategy. Results are discussed in terms of cognitive models of OC symptoms and their implications for cognitive-behavioral therapy.     

Need for cognitive closure and coping strategies

This study investigated the hypothesis that the process of coping may be motivated by an interaction of directional motivational factors represented by job satisfaction/dissatisfaction and by non‐directional or epistemological motivational factors represented by the level of Need for Cognitive Closure. Need for Cognitive Closure is based on two general tendencies: the urgency tendency (“to seize”) and the permanency tendency (“to freeze”). The urgency tendency reflects the desire to attain closure as quickly as possible (“to seize”) whereas the permanence tendency refers to the inclination to maintain closure for as long as possible along with a desire to preserve or “freeze on” past knowledge and to safeguard future knowledge. The urgency and permanence notions both rest on the assumption that people under a heightened need for closure experience its absence as aversive. In relation with this, we predicted that if job satisfaction is low, the increased need for closure is related to the choice of problem‐oriented coping strategies. Alternatively, we hypothesized that with high job satisfaction an increased need for closure is related to use of avoidance coping. Questionnaires pertaining to need for cognitive closure, to coping strategies and to a measure of job satisfaction were completed by a group of 146 Croatian immigrants living in Italy. Results of the analyses confirmed that when subjects were highly satisfied with their job, their primary concern was to preserve their position. So, here the high need for closure enhanced the tendency to freeze and induced the choice of avoidance coping strategies. On the other hand, our results confirmed that when persons are not satisfied with their job, high need for closure increases the desire for change and improvement. Immigrants increased the tendency to seize manifests itself in extensive and quickened information processing relative to the use of problem‐oriented coping strategies.     

Academic goals, cognitive and self-regulatory strategies

In this paper we analyze differences between cognitive (selection, organizational, elaboration and rehearsal) and self-regulatory (planning, monitoring-regulating) strategies depending on academic goals. A total of 447 obligatory secondary education students (55% boys and 44,7% girls) participated in this study. The results indicated that as well as learning goals, other goals (obtain social approval or a good job) are related to the use of strategies and study engagement. Even performance-approach and performance-avoidance goals are significantly related to cognitive and self-regulatory strategies use.     

How language acquisition builds on cognitive development

When children acquire a first language, they build on what they know--conceptual information that discriminates and helps create categories for the objects, relations and events they experience. This provides the starting point for language from the age of 12 months on. So children first set up conceptual representations, then add linguistic representations for talking about experience. Do they then discard earlier conceptual representations in favour of linguistic ones, or do they retain them? Recent research on the coping strategies that young children (and adults) rely on when they are unable to draw on language suggest that they retain both types of representations for use as needed.     

路径知识习得的认知老化效应

年龄与路径知识习得的关系是空间与认知老化两大领域研究的重要议题。老年人在面对不同的路径学习环境时呈现出不同的认知老化表现。以往与年龄相关的路径知识习得能力变化的研究, 主要支持了认知老化衰退理论。然而近来发现随着年龄增长, 老年人保留了一种空间认知补偿能力。由此, 在对前人文献进行回顾和反思的基础上, 整合路径知识习得的认知老化表现及机制以探究缓解空间认知老化可能的内部因素和外部有效措施。     

Age-related differences in the course of cognitive skill acquisition: the role of regional cortical shrinkage and cognitive resources

This study examined the impact of age-related differences in regional cerebral volumes and cognitive resources on acquisition of a cognitive skill. Volumes of brain regions were measured on magnetic resonance images of healthy adults (aged 22-80). At the early stage of learning to solve the Tower of Hanoi puzzle, speed and efficiency were associated with age, prefrontal cortex volume, and working memory. A similar pattern of brain-behavior associations was observed with perseveration measured on the Wisconsin Card Sorting Test. None of the examined structural brain variables were important at the later stages of skill acquisition. When hypertensive participants were excluded, the effect of prefrontal shrinkage on executive aspects of performance was no longer significant, but the effect of working memory remained.     

The role of phonological and executive working memory resources in simple arithmetic strategies

The current study investigated the role of the central executive and the phonological loop in arithmetic strategies to solve simple addition problems (Experiment 1) and simple subtraction problems (Experiment 2). The choice/no-choice method was used to investigate strategy execution and strategy selection independently. The central executive was involved in both retrieval and procedural strategies, but played a larger role in the latter than in the former. Active phonological processes played a role in procedural strategies only. Passive phonological resources, finally, were only needed when counting was used to solve subtraction problems. No effects of working memory load on strategy selection were observed.