Cognitive arithmetic: comparison of operations

Adults' performance of simple arithmetic calculations (addition, multiplication, and numerical comparison) was examined to test predictions of digital (counting), analog, and network models. Although all of these models have been supported by studies of mental addition, each leads to a different prediction concerning relations between the times required for addition, multiplication, and numerical comparison. Pairs of single-digit integers were presented and reaction times (RTs) for adding, multiplying, and comparing the stimuli were collected. A high correlation between RT for addition and multiplication of the same digits was obtained. This result is consistent with a network model, but presents difficulties for both analog and counting models. A "ties" effect of no increase in RT with increases in problem size for doubles such as 2 + 2 has been found in previous studies of addition using verification procedures, but was not found with the production task employed in the present study. Instead, a different kind of ties effect was found. Reaction time for both addition and multiplication of ties increased more slowly with problem size than did RT for non-tie problems. This ties effect, and the finding that probability of making errors contributes independently of problem size to RT support a distinction between location and accessibility of information in a network.     

The strategic use of alternative representations in arithmetic word problem solving

Multiple-step arithmetic problems can be solved by diverse strategies depending on the mental representation constructed by individuals from the situation described in the text of the problem. This representation will indeed determine the organization of subgoals to be reached or in other words the order of completion of calculations. This study aims at determining the conditions under which specific strategies are set up by adults. Using a paradigm that allows us to assess when calculations are performed, we show that adults usually organize their subgoals as they are explicitly mentioned in the problem, even though a strategy that is less demanding on working memory could have been used. However, we show that increasing the difficulty of the problem leads individuals to set up more economic strategies. Moreover, these economic strategies are even more likely to be used when the cognitive cost of the construction of the representation they rely on is low.     

Introduction: Internal representations and parent-infant relationships

Infant mental health practitioners have continually striven to understand and to improve disordered parent-infant relationships. Recently, researchers have begun attempts to describe and study these relationships systematically. There is an increasing interest in the construction of internal representations as indices of how relationship patterns are carried internally and transmitted from parent to child. This special issue of the Infant Mental Health Journal presents theoretical and clinical perspectives on internal representations, as well as results from some preliminary empirical investigations. This new convergence of efforts among theoreticians, clinicians, and investigators may yield important advances in our efforts to capture the complexity and riches of parent-infant relationships and their disorders.     

Efficiency of arithmetic operations in bilinguals as a function of language

English-Spanish bilinguals solved simple arithmetic problems and were required to respond In their preferred (P) language. the language In which they originally learned arithmetic, or In their nonpreferred (NP) language. Each arithmetic problem required one. two, or three addition operations. Reaction time was a linear function of number of operations. The intercept for the P language was lower than that for the NP language. but there were no differences In slope. The intercept difference was interpreted In terms of translation time. either as translation of the sum from the P to the NP language or as translation from an abstract representation to the NP as opposed to the P language.     

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.     

Exposing operations of power in supervisory relationships

Through a poststructural lens, we examine how power may show itself in relationships between supervisees and supervisors, producing both helpful and harmful effects. Drawing from our own experiences, as well as conversations with other members of our supervisory group, we demonstrate how privileged discourses around professional status, gender, and race may bring about difficulties including a sense of doubt, worry, inadequacy, and a fear of speaking up. We also illustrate how these difficulties can be addressed in a manner that may lessen their influence, while increasing supervisees' sense of agency.     

Cognitive representations and processes in arithmetic: inferences from the performance of brain-damaged subjects.

In this article, we present data from two brain-damaged patients with calculation impairments in support of claims about the cognitive mechanisms underlying simple arithmetic performance. We first present a model of the functional architecture of the cognitive calculation system based on previous research. We then elaborate this architecture through detailed examination of the patterns of spared and impaired performance of the two patients. From the patients' performance we make the following theoretical claims: that some arithmetic facts are stored in the form of individual fact representations (e.g., 9 x 4 = 36), whereas other facts are stored in the form of a general rule (e.g., 0 x N = 0); that arithmetic fact retrieval is mediated by abstract internal representations that are independent of the form in which problems are presented or responses are given; that arithmetic facts and calculation procedures are functionally independent; and that calculation algorithms may include special-case procedures that function to increase the speed or efficiency of problem solving. We conclude with a discussion of several more general issues relevant to the reported research.     

Cognitive representations of functional interactions with objects

Two studies addressed people’s knowledge about the movements underlying functional interactions with objects, when the interactions were described by simple verbal labels expressing environmental goals. In Experiment 1, subjects rated each action with respect to six dimensions: which portion of the limb moved, distance moved, forcefulness, effectors involved, size of the contact surface, and resemblance to grasp. Ratings were systematic and fell on two distinct underlying factors related to limb movement and effector (usually the hand) configuration. In Experiment 2, subjects sorted a subset of the actions by similarity of movement. Clustering and multidimensional scaling solutions indicated that the six initial dimensions contributed to similarity judgments, along with additional parameters. The results support the existence of cognitively accessible, but still relatively specific, representations of functional actions, with potential implications for motor and memory performance.     

Mental representations of arithmetic facts: evidence from eye movement recordings supports the preferred operand-order-specific representation hypothesis

There are three main hypotheses about mental representations of arithmetic facts: the independent representation hypothesis, the operand-order-free single-representation hypothesis, and the operand-order-specific single-representation hypothesis. The current study used electrical recordings of eye movements to examine the organization of arithmetic facts in long-term memory. Subjects were presented single-digit addition and multiplication problems and were asked to report the solutions. Analyses of the horizontal electrooculograph (HEOG) showed an operand order effect for multiplication in the time windows 150-300?ms (larger negative potentials for smaller operand first problems than for larger operand first ones). The operand order effect was reversed in the time windows from 400 to 1,000?ms (i.e., larger operand first problems had larger negative potentials than smaller operand first problems). For addition, larger operand first problems had larger negative potentials than smaller operand first in the series of time windows from 300 to 1,000?ms, but the effect was smaller than that for multiplication. These results confirmed the dissociated representation of addition and multiplication facts and were consistent with the prediction of the preferred operand-order-specific representation hypothesis.     

Mental representations of arithmetic facts: Evidence from eye movement recordings supports the preferred operand-order-specific representation hypothesis

There are three main hypotheses about mental representations of arithmetic facts: the independent representation hypothesis, the operand-order-free single-representation hypothesis, and the operand-order-specific single-representation hypothesis. The current study used electrical recordings of eye movements to examine the organization of arithmetic facts in long-term memory. Subjects were presented single-digit addition and multiplication problems and were asked to report the solutions. Analyses of the horizontal electrooculograph (HEOG) showed an operand order effect for multiplication in the time windows 150–300 ms (larger negative potentials for smaller operand first problems than for larger operand first ones). The operand order effect was reversed in the time windows from 400 to 1,000 ms (i.e., larger operand first problems had larger negative potentials than smaller operand first problems). For addition, larger operand first problems had larger negative potentials than smaller operand first in the series of time windows from 300 to 1,000 ms, but the effect was smaller than that for multiplication. These results confirmed the dissociated representation of addition and multiplication facts and were consistent with the prediction of the preferred operand-order-specific representation hypothesis.     

The logic of arithmetic

Conclusion The arithmetic developed along the lines indicated by Cantor has up until now found no application in the real, as opposed to the mathematical, world. It is to be hoped that the changes suggested in this paper bring mathematics closer to human thought and allow it to be of increasing benefit in its service to mankind.     

Primacy of functional knowledge in semantic representations: The case of living and nonliving things

In 3 experiments, participants decided whether sensory and functional features were true of living and nonliving concepts. In Experiments 1 and 2, concepts were presented twice: test phase followed study phase after either 3 min (Experiment 1) or 3 s (Experiment 2). At test, concepts were paired with the same feature as that at study, or a different feature from either the same modality (within-modality priming) or another modality (cross-modality priming). In both experiments functional decisions were faster than sensory decisions for living and nonliving concepts. Whilst no semantic priming occurred between study and test in Experiment 1, the shorter study–test interval of Experiment 2 did lead to test phase semantic priming. Here there was greater within- than cross-modality priming for sensory decisions, but equivalent within- and cross-modality priming for functional decisions owing to significantly greater facilitation of functional decisions from prior sensory decisions than vice versa. Experiment 3 involved a single verification phase: For half the participants the feature name preceded the concept name, and for half the concept name preceded the feature name. The functional processing advantage persisted irrespective of presentation order. Results suggest that functional information is central to the representation of all concepts: Function is processed faster than sensory information and is activated obligatorily.     

Parental socioeconomic status and the neural basis of arithmetic: differential relations to verbal and visuo‐spatial representations

We examined the relation of parental socioeconomic status (SES) to the neural bases of subtraction in school‐age children (9‐ to 12‐year‐olds). We independently localized brain regions subserving verbal versus visuo‐spatial representations to determine whether the parental SES‐related differences in children's reliance on these neural representations vary as a function of math skill. At higher SES levels, higher skill was associated with greater recruitment of the left temporal cortex, identified by the verbal localizer. At lower SES levels, higher skill was associated with greater recruitment of right parietal cortex, identified by the visuo‐spatial localizer. This suggests that depending on parental SES, children engage different neural systems to solve subtraction problems. Furthermore, SES was related to the activation in the left temporal and frontal cortex during the independent verbal localizer task, but it was not related to activation during the independent visuo‐spatial localizer task. Differences in activation during the verbal localizer task in turn were related to differences in activation during the subtraction task in right parietal cortex. The relation was stronger at lower SES levels. This result suggests that SES‐related differences in the visuo‐spatial regions during subtraction might be based in SES‐related verbal differences.     

Massive redeployment,exaptation, and the functional integration of cognitive operations

The massive redeployment hypothesis (MRH) is a theory about the functional topography of the human brain, offering a middle course between strict localization on the one hand, and holism on the other. Central to MRH is the claim that cognitive evolution proceeded in a way analogous to component reuse in software engineering, whereby existing components—originally developed to serve some specific purpose—were used for new purposes and combined to support new capacities, without disrupting their participation in existing programs. If the evolution of cognition was indeed driven by such exaptation, then we should be able to make some specific empirical predictions regarding the resulting functional topography of the brain. This essay discusses three such predictions, and some of the evidence supporting them. Then, using this account as a background, the essay considers the implications of these findings for an account of the functional integration of cognitive operations. For instance, MRH suggests that in order to determine the functional role of a given brain area it is necessary to consider its participation across multiple task categories, and not just focus on one, as has been the typical practice in cognitive neuroscience. This change of methodology will motivate (even perhaps necessitate) the development of a new, domain-neutral vocabulary for characterizing the contribution of individual brain areas to larger functional complexes, and direct particular attention to the question of how these various area roles are integrated and coordinated to result in the observed cognitive effect. Finally, the details of the mix of cognitive functions a given area supports should tell us something interesting not just about the likely computational role of that area, but about the nature of and relations between the cognitive functions themselves. For instance, growing evidence of the role of “motor” areas like M1, SMA and PMC in language processing, and of “language” areas like Broca’s area in motor control, offers the possibility for significantly reconceptualizing the nature both of language and of motor control.     

Combined antecedent variables as motivating operations within functional analyses

Functional analysis test conditions typically manipulate a single antecedent variable and an associated consequence to better isolate response-reinforcer relations. In some instances no problem behavior is observed, perhaps representing a false-negative finding. The present study evaluated one approach to assess potentially false-negative findings within functional analyses. Participants were exposed to single-antecedent functional analysis test conditions and combined-antecedent test conditions within a multielement design. Both participants engaged in problem behavior primarily during the combined-antecedent test conditions, and treatments matched to the results were effective in reducing problem behavior. Findings are discussed in terms of clinical implications of combining antecedent variables to further examine potentially false-negative functional analysis results.     

The relationships between functional and dysfunctional impulsivity and aggression across different samples

E. S. Barratt proposed the term impulsive aggression to define a kind of aggression that is characterized by acting without thinking because of high levels of impulsivity. Previous research using psychometric measures has shown that impulsivity and aggression are related as far as psychometric measures are concerned. Nevertheless, most of the research has been done with samples of university students. Our research tests whether this relationship is stable across different samples; university students, teenagers and workers. Our results show that impulsivity and aggression have a consistent pattern of relationships across these samples, with impulsivity being specially related to emotional and instrumental aspects of aggression. Furthermore, the effects of anger on aggression seem to show a pattern of relationship that depends on age, with a tendency to physical aggression in young people and verbal aggression in adults.     

Children's representations of multiple family relationships: organizational structure and development in early childhood

The authors examine mutual family influence processes at the level of children's representations of multiple family relationships, as well as the structure of those representations. From a community sample with 3 waves, each spaced 1 year apart, kindergarten-age children (105 boys and 127 girls) completed a story-stem completion task, tapping representations of multiple family relationships. Structural equation modeling with autoregressive controls indicated that representational processes involving different family relationships were interrelated over time, including links between children's representations of marital conflict and reactions to conflict, between representations of security about marital conflict and parent-child relationships, and between representations of security in father-child and mother-child relationships. Mixed support was found for notions of increasing stability in representations during this developmental period. Results are discussed in terms of notions of transactional family dynamics, including family-wide perspectives on mutual influence processes attributable to multiple family relationships.     

The non-abstractness of mental representations

An argument is made that formal abstract mental representations are lacking because they fail to accommodate the contribution of the medium. The importance of the medium is singled out and several respects in which it resists formal characterization noted. Specifically, it is pointed out that the medium can always be relevant, that the aspects by which it is made so are not in principle demarcated or constrained, and that they cannot be restricted a priori at the time of encoding. The examination of several ways by which specifications of medium could be incorporated into mental representations suggests that to obtain a full account of the medium one has to abandon some basic characteristics of mental representations. Conversely, maintaining the concept of mental representation in its standard sense leaves the medium, and with it some fundamental properties of human cognition, outside of one's scientific account. This conclusion serves as the basis for more general comments regarding the study of mind.     

Primacy of functional knowledge in semantic representations: the case of living and nonliving things

In 3 experiments, participants decided whether sensory and functional features were true of living and nonliving concepts. In Experiments 1 and 2, concepts were presented twice: test phase followed study phase after either 3 min (Experiment 1) or 3 s (Experiment 2). At test, concepts were paired with the same feature as that at study, or a different feature from either the same modality (within-modality priming) or another modality (cross-modality priming). In both experiments functional decisions were faster than sensory decisions for living and nonliving concepts. Whilst no semantic priming occurred between study and test in Experiment 1, the shorter study-test interval of Experiment 2 did lead to test phase semantic priming. Here there was greater within- than cross-modality priming for sensory decisions, but equivalent within- and cross-modality priming for functional decisions owing to significantly greater facilitation of functional decisions from prior sensory decisions than vice versa. Experiment 3 involved a single verification phase: For half the participants the feature name preceded the concept name, and for half the concept name preceded the feature name. The functional processing advantage persisted irrespective of presentation order. Results suggest that functional information is central to the representation of all concepts: Function is processed faster than sensory information and is activated obligatorily.