On the development of the mental number line: more, less, or never holistic with increasing age?

Magnitude is assumed to be represented along a holistic mental number line in adults. However, the authors recently observed a unit-decade compatibility effect for 2-digit numbers that is inconsistent with this "holisticness" assumption (H.-C. Nuerk, U. Weger, & K. Willmes, 2001). This study used the compatibility effect to examine whether the mental number line representation of magnitude changes toward greater or less holisticness in children from Grades 2-5. The results indicate that decades and units of 2-digit numbers are processed separately rather than holistically from Grade 2 on. However, this separate processing seems to develop from a more sequential (left-to-right) to a more parallel processing mode. Moreover, children may use different strategies depending on task demands. The results are interpreted in the framework of Siegler's overlapping waves model.     

Task instructions determine the visuospatial and verbal–spatial nature of number–space associations

Evidence for number–space associations comes from the spatial–numerical association of response codes (SNARC) effect, consisting in faster reaction times to small/large digits with the left/right hand, respectively. Two different proposals are commonly discussed concerning the cognitive origin of the SNARC effect: the visuospatial account and the verbal–spatial account. Recent studies have provided evidence for the relative dominance of verbal–spatial over visuospatial coding mechanisms, when both mechanisms were directly contrasted in a magnitude comparison task. However, in these studies, participants were potentially biased towards verbal–spatial number processing by task instructions based on verbal–spatial labels. To overcome this confound and to investigate whether verbal–spatial coding mechanisms are predominantly activated irrespective of task instructions, we completed the previously used paradigm by adding a spatial instruction condition. In line with earlier findings, we could confirm the predominance of verbal–spatial number coding under verbal task instructions. However, in the spatial instruction condition, both verbal–spatial and visuospatial mechanisms were activated to an equal extent. Hence, these findings clearly indicate that the cognitive origin of number–space associations does not always predominantly rely on verbal–spatial processing mechanisms, but that the spatial code associated with numbers is context dependent.     

Children’s early mental number line: Logarithmic or decomposed linear?

Recently, the nature of children’s mental number line has received much investigation. In the number line task, children are required to mark a presented number on a physical number line with fixed endpoints. Typically, it was observed that the estimations of younger/inexperienced children were accounted for best by a logarithmic function, whereas those of older/more experienced children were reflected best by a linear function. This led to the conclusion that children’s mental number line transforms from logarithmic to linear with age and experience. In this study, we outline an alternative interpretation of children’s performance in a number line task. We suggest that two separate linear representations for one- and two-digit numbers may exist in young children and that initially the integration of these two representations into the place value structure of the Arabic number system is not fully mastered. When testing this assumption in a sample of more than 120 first graders, we observed that the two-linear model consistently provided better fit indexes. We conclude that instead of assuming a transition from logarithmic to linear coding, performance differences could also be accounted for by an improvement in integrating tens and units into the Arabic place value system.     

A neural model of hippocampal–striatal interactions in associative learning and transfer generalization in various neurological and psychiatric patients

Building on our previous neurocomputational models of basal ganglia and hippocampal region function (and their modulation by dopamine and acetylcholine, respectively), we show here how an integration of these models can inform our understanding of the interaction between the basal ganglia and hippocampal region in associative learning and transfer generalization across various patient populations. As a common test bed for exploring interactions between these brain regions and neuromodulators, we focus on the acquired equivalence task, an associative learning paradigm in which stimuli that have been associated with the same outcome acquire a functional similarity such that subsequent generalization between these stimuli increases. This task has been used to test cognitive dysfunction in various patient populations with damages to the hippocampal region and basal ganglia, including studies of patients with Parkinson’s disease (PD), schizophrenia, basal forebrain amnesia, and hippocampal atrophy. Simulation results show that damage to the hippocampal region—as in patients with hippocampal atrophy (HA), hypoxia, mild Alzheimer’s (AD), or schizophrenia—leads to intact associative learning but impaired transfer generalization performance. Moreover, the model demonstrates how PD and anterior communicating artery (ACoA) aneurysm—two very different brain disorders that affect different neural mechanisms—can have similar effects on acquired equivalence performance. In particular, the model shows that simulating a loss of dopamine function in the basal ganglia module (as in PD) leads to slow acquisition learning but intact transfer generalization. Similarly, the model shows that simulating the loss of acetylcholine in the hippocampal region (as in ACoA aneurysm) also results in slower acquisition learning. We argue from this that changes in associative learning of stimulus–action pathways (in the basal ganglia) or changes in the learning of stimulus representations (in the hippocampal region) can have similar functional effects.     

Larger,smaller, odd or even? Task-specific effects of optokinetic stimulation on the mental number space

Previous studies have shown that number processing can induce spatial biases in perception and action and can trigger the orienting of visuospatial attention. Few studies, however, have investigated how spatial processing and visuospatial attention influences number processing. In the present study, we used the optokinetic stimulation (OKS) technique to trigger eye movements and thus overt orienting of visuospatial attention. Participants were asked to stare at OKS, while performing parity judgements (Experiment 1) or number comparison (Experiment 2), two numerical tasks that differ in terms of demands on magnitude processing. Numerical stimuli were acoustically presented, and participants responded orally. We examined the effects of OKS direction (leftward or rightward) on number processing. The results showed that rightward OKS abolished the classic number size effect (i.e., faster reaction times for small than large numbers) in the comparison task, whereas the parity task was unaffected by OKS direction. The effect of OKS highlights a link between visuospatial orienting and processing of number magnitude that is complementary to the more established link between numerical and visuospatial processing. We suggest that the bidirectional link between numbers and space is embodied in the mechanisms subserving sensorimotor transformations for the control of eye movements and spatial attention.     

Exploring the dynamics of the appraisal–emotion relationship: A constraint satisfaction model of the appraisal process

This paper presents a computational model of emotions that is based in an appraisal theoretical framework. The model explores the dynamics of the appraisal–emotion relationship using parallel constraint satisfaction. It proposes belief coherence, desire coherence and belief–desire coherence as basic principles and specifies how these principles can be realised in a parallel constraint satisfaction system. The model shows that the very same mechanisms that account for the formation of coherence or consistency are also pivotal for the emergence and alteration of emotional states and for the dynamics of the cognition–emotion relationship. Concrete simulation examples show how discrete emotions such as anger, pride, sadness, shame, surprise, relief and disappointment emerge out of these principles.     

Decade breaks in the mental number line? Putting the tens and units back in different bins.

Most models of number recognition agree that among other number representations there is a central semantic magnitude representation which may be conceptualized as a logarithmically compressed mental number line. Whether or not this number line is decomposed into different representations for tens and units is, however, controversial. We investigated this issue in German participants in a magnitude comparison (selection) task in which the larger of two visually presented Arabic two-digit numbers had to be selected. Most importantly, we varied unit-decade-compatibility: a number pair was defined as compatible if the decade magnitude comparison and the unit magnitude comparison of the two numbers would lead to the same response (e.g. 52 and 67) and as incompatible if this was not the case (e.g. 47 and 62). While controlling for overall numerical distance, size and other variables, we consistently found compatibility effects. A control experiment showed that this compatibility effect was not due to perceptual presentation characteristics. We conclude that the idea of one single number line representation that does not additionally assume different magnitude representations for tens and units is not sufficient to account for the data. Finally, we discuss why decade effects were not found in other experimental settings.     

The impact of inhibition capacities and age on number–space associations

Numerical and spatial representations are tightly linked, i.e., when doing a binary classification judgment on Arabic digits, participants are faster to respond with their left/right hand to small/large numbers, respectively (Spatial-Numerical Association of Response Codes, SNARC effect, Dehaene et al. in J Exp Psychol Gen 122:371–396, 1993). To understand the underlying mechanisms of the well-established SNARC effect, it seems essential to explore the considerable inter-individual variability characterizing it. The present study assesses the respective roles of inhibition, age, working memory (WM) and response speed. Whereas these non-numerical factors have been proposed as potentially important factors to explain individual differences in SNARC effects, none (except response speed) has so far been explored directly. Confirming our hypotheses, the results show that the SNARC effect was stronger in participants that had weaker inhibition abilities (as assessed by the Stroop task), were relatively older and had longer response times. Interestingly, whereas a significant part of the age influence was mediated by cognitive inhibition, age also directly impacted the SNARC effect. Similarly, cognitive inhibition abilities explained inter-individual variability in number–space associations over and above the factors age, WM capacity and response speed. Taken together our results provide new insights into the nature of number–space associations by describing how these are influenced by the non-numerical factors age and inhibition.     

Approximate number sense,symbolic number processing,or number–space mappings: What underlies mathematics achievement?

In this study, the performance of typically developing 6- to 8-year-old children on an approximate number discrimination task, a symbolic comparison task, and a symbolic and nonsymbolic number line estimation task was examined. For the first time, children’s performances on these basic cognitive number processing tasks were explicitly contrasted to investigate which of them is the best predictor of their future mathematical abilities. Math achievement was measured with a timed arithmetic test and with a general curriculum-based math test to address the additional question of whether the predictive association between the basic numerical abilities and mathematics achievement is dependent on which math test is used. Results revealed that performance on both mathematics achievement tests was best predicted by how well children compared digits. In addition, an association between performance on the symbolic number line estimation task and math achievement scores for the general curriculum-based math test measuring a broader spectrum of skills was found. Together, these results emphasize the importance of learning experiences with symbols for later math abilities.     

The neural substrate for dreaming: is it a subsystem of the default network?

Building on the content, developmental, and neurological evidence that there are numerous parallels between waking cognition and dreaming, this article argues that the likely neural substrate that supports dreaming, which was discovered through converging lesion and neuroimaging studies, may be a subsystem of the waking default network, which is active during mind wandering, daydreaming, and simulation. Support for this hypothesis would strengthen the case for a more general neurocognitive theory of dreaming that starts with established findings and concepts derived from studies of waking cognition and neurocognition. If this theory is correct, then dreaming may be the quintessential cognitive simulation because it is often highly complex, often includes a vivid sensory environment, unfolds over a duration of a few minutes to a half hour, and is usually experienced as real while it is happening.     

Size congruity effects with two-digit numbers: Expanding the number line?

The size congruity effect is the interesting result that comparisons of the sizes of the physical formats in which numerals appear are affected by the numerical magnitudes of the respective numerals. We demonstrated that separating the physical and the numerical attributes in space leaves the effect unchanged. We then applied the spatially separated version to two-digit numerals and showed the effect to be comparable to that obtained with single numerals. We showed further that the effect is sensitive to the relative salience of the numeric and physical dimensions, to the extent that when the latter is the more salient dimension, a reverse effect obtains by which physical size interferes with number comparison. The results can be explained by a relative speed of processing account, but they are also compatible with an attention account that does not appeal to the notion of automaticity.     

Comparing implicit and synaesthetic number–space associations: Visuospatial and verbal spatial–numerical associations of response codes

In the theory of the mental number line, number, and space are implicitly associated, giving rise to the spatial–numerical association of response codes (SNARC) effect, in which smaller numbers are more readily associated with the left side of space and larger numbers with the right, during a parity judgement task. Others, however, have argued that the SNARC effect is flexible and better explained by verbal rather than spatial associations. A few single-case studies on the SNARC effect have tested number–space synaesthetes, who make explicit associations between number and space. Here, we present data from experiments conducted on groups of synaesthetes and nonsynaesthetes on the classic SNARC parity judgement task with lateralized response keys and a modified version in which they responded to labels appearing on screen. Synaesthetes' behaviour was expected to differ from nonsynaesthetes' behaviour due to the explicit, fixed nature of their number–space associations, but both experiments show the two groups behaving in the same way, indicating that parity judgement tasks may not be tapping the same representation of number that gives rise to synaesthetic number–space experiences.     

A colorful walk, but is it on the mental number line? Reply to Cohen Kadosh, Tzelgov, and Henik

Cohen Kadosh, Tzelgov, and Henik [Cohen Kadosh, R., Tzelgov, J., and Henik, A. (2008). A synesthetic walk on the number line: The size effect. Cognition, 106, 548-557] present a new paradigm to probe properties of the mental number line. They describe two experiments which they argue to be inconsistent with the exact small number model proposed by Verguts, Fias, and Stevens [Verguts, T., Fias, W., Stevens, M. (2005). A model of exact small-number representation. Psychonomic Bulletin and Review, 12, 66-80]. We discuss the data, assumptions, and conclusions of Cohen Kadosh et al.'s paper in relation to existing models of numerical cognition.     

Community-delivered infant–parent psychotherapy improves maternal sensitive caregiving: Evaluation of the Michigan model of infant mental health home visiting

The current study evaluated the effectiveness of a home-based psychotherapeutic Infant Mental Health Home Visiting (IMH-HV) intervention for enhancing parenting sensitivity; a secondary aim was to evaluate whether the use of video feedback was associated with greater treatment response. Participants were N = 78 mothers and their children (age at entry ranged from prebirth to 24-month old (M = 9.8, SD = 8.4), who were initiating IMH-HV services with community mental health-based therapists (N = 51). Dyads were assessed during extended home visits via standardized interviews and observational and questionnaire methods within the first month of treatment (baseline), and again 6 and 12 months thereafter. Following each of these extended home visits, study evaluators completed a standard Q-sort to capture observations of maternal sensitivity during the visit. Therapists completed fidelity checklists used to derive the total number of IMH-HV sessions received (i.e., dosage) and frequency with which therapists provided video feedback. Results indicated a dose–response relationship between number of sessions and maternal sensitivity, and that video review with parents independently contributed to improved maternal sensitivity. Discussion focuses on the effectiveness of this community-based psychotherapeutic home visiting model for enhancing parenting, as well as the value of video feedback as a specific therapeutic strategy.     

Reconceptualizing the origins of number knowledge: A “non-numerical” account

This paper presents a new conceptualization of the origins of numerical competence in humans. I first examine the existing claim that infants are innately provided with a system of specifically numerical knowledge, consisting of both cardinal and ordinal concepts. I suggest instead that the observed behaviors require only simple perceptual discriminations based on domain-independent competencies. At most, these involve the formal equivalent of cardinal information. Finally, I present a “non-numerical” account that characterizes infants competencies with regard to numerosity as emerging primarily from some general characteristics of the human perception and attention system.     

How psychological stress in the workplace influences presenteeism propensity: A test of the Demand–Control–Support model

The aim of this study is to examine the influence of the sources of psychological stress at work assumed by the Demand–Control–Support (DCS) model on presenteeism propensity. Research propositions are derived from the dynamic version of the Demand–Control model, to which we added the dimension of social support, which is part of the DCS model. Data were obtained from a large representative sample consisting of 1609 public and private sector workers living in the Canadian province of Québec, who responded to a questionnaire, administered using computer-assisted interviewing over a 3-month period in 2007–2008. The results indicate that the combination of low decision authority and high supervisor's support attenuates the positive relationship between job demands and presenteeism propensity within a 10-year time frame of exposure to a new job.     

Beyond the “hot-and-cold” game: A demonstration of computer-controlled shaping

Shaping, or the method of successive approximations, is widely taught in introductory psychology and the psychology of learning as a procedure for establishing new behavior. This article illustrates a computer-controlled shaping demonstration that allows the user to specify several critical parameters of the shaping process and that then shapes the user’s mouse movements toward an arbitrary virtual (invisible) target on the computer screen. The relative effectiveness of different shaping parameters can be assessed by examining several dependent measures, such as the distance of the cursor from the target across time and the rate at which reinforcers were earned. This demonstration allows students to move beyond the notion that shaping is simply the application of the “hot-and-cold” game and to understand that there is a science underlying the art of shaping.     

Coding strategies in number space: Memory requirements influence spatial–numerical associations

The tendency to respond faster with the left hand to relatively small numbers and faster with the right hand to relatively large numbers (spatial numerical association of response codes, SNARC effect) has been interpreted as an automatic association of spatial and numerical information. We investigated in two experiments the impact of task-irrelevant memory representations on this effect. Participants memorized three Arabic digits describing a left-to-right ascending number sequence (e.g., 3–4–5), a descending sequence (e.g., 5–4–3), or a disordered sequence (e.g., 5–3–4) and indicated afterwards the parity status of a centrally presented digit (i.e., 1, 2, 8, or 9) with a left/right keypress response. As indicated by the reaction times, the SNARC effect in the parity task was mediated by the coding requirements of the memory tasks. That is, a SNARC effect was only present after memorizing ascending or disordered number sequences but disappeared after processing descending sequences. Interestingly, the effects of the second task were only present if all sequences within one experimental block had the same type of order. Taken together, our findings are inconsistent with the idea that spatial–numerical associations are the result of an automatic and obligatory cognitive process but do suggest that coding strategies might be responsible for the cognitive link between numbers and space.     

Reconstructing the Risk–Need–Responsivity model: A theoretical elaboration and evaluation

In this paper, we examine the theoretical strengths and weaknesses of the Risk–Need–Responsivity (RNR) model of offender rehabilitation. We briefly discuss the nature of rehabilitation theories and their core components and then review the three source theories associated with the RNR Model. Following this we set out to reconstruct the RNR model in light of this analysis, essentially arguing that there are at least three components to any rehabilitation theory: (a) primary aims, values and principles; (b) etiological and methodological assumptions; and (c) practice implications. We then evaluate the theoretical and empirical adequacy of the RNR model. Finally, we conclude the paper with a few comments on the policy, research, and clinical implications of our evaluation (and reconstruction) of this important rehabilitation model.