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.     

Sequential or parallel decomposed processing of two-digit numbers? Evidence from eye-tracking

While reaction time data have shown that decomposed processing of two-digit numbers occurs, there is little evidence about how decomposed processing functions. Poltrock and Schwartz (1984) argued that multi-digit numbers are compared in a sequential digit-by-digit fashion starting at the leftmost digit pair. In contrast, Nuerk and Willmes (2005) favoured parallel processing of the digits constituting a number. These models (i.e., sequential decomposition, parallel decomposition) make different predictions regarding the fixation pattern in a two-digit number magnitude comparison task and can therefore be differentiated by eye fixation data. We tested these models by evaluating participants' eye fixation behaviour while selecting the larger of two numbers. The stimulus set consisted of within-decade comparisons (e.g., 53_57) and between-decade comparisons (e.g., 42_57). The between-decade comparisons were further divided into compatible and incompatible trials (cf. Nuerk, Weger, & Willmes, 2001) and trials with different decade and unit distances. The observed fixation pattern implies that the comparison of two-digit numbers is not executed by sequentially comparing decade and unit digits as proposed by Poltrock and Schwartz (1984) but rather in a decomposed but parallel fashion. Moreover, the present fixation data provide first evidence that digit processing in multi-digit numbers is not a pure bottom-up effect, but is also influenced by top-down factors. Finally, implications for multi-digit number processing beyond the range of two-digit numbers are discussed.     

What makes multiplication facts difficult. Problem size or neighborhood consistency?

Two current models of arithmetic fact retrieval, the network interference theory (NIT; Campbell, 1995) and the interacting neighbors (IN) model (Verguts & Fias, 2005a), predict that errors in simple multiplication should be more probable, if they include the same digit as the correct result (i.e., if they are "consistent," compared with "inconsistent" errors). In a reanalysis of error data originally reported by Campbell (1997), we provide first empirical evidence for this prediction. Furthermore, these results support the notion of different quantity representations for decades and units as proposed by Nuerk, Weger, and Willmes (2001). However, the NIT and IN-model differ in their explanations of the problem-size effect, a hallmark finding robustly observed in arithmetic fact retrieval. Only the IN-model predicts that a correct answer's neighborhood consistency can fully account for the problem-size effect, which was confirmed in our analysis.     

Multi-digit number processing beyond the two-digit number range: a combination of sequential and parallel processes

Investigations of multi-digit number processing typically focus on two-digit numbers. Here, we aim to investigate the generality of results from two-digit numbers for four- and six-digit numbers. Previous studies on two-digit numbers mostly suggested a parallel processing of tens and units. In contrast, the few studies examining the processing of larger numbers suggest sequential processing of the individual constituting digits. In this study, we combined the methodological approaches of studies implying either parallel or sequential processing. Participants completed a number magnitude comparison task on two-, four-, and six-digit numbers including unit-decade compatible and incompatible differing digit pairs (e.g., 32_47, 3<4 and 2<7 vs. 37_52, 3<5 but 7>2, respectively) at all possible digit positions. Response latencies and fixation behavior indicated that sequential and parallel decomposition is not exclusive in multi-digit number processing. Instead, our results clearly suggested that sequential and parallel processing strategies seem to be combined when processing multi-digit numbers beyond the two-digit number range. To account for the results, we propose a chunking hypothesis claiming that multi-digit numbers are separated into chunks of shorter digit strings. While the different chunks are processed sequentially digits within these chunks are processed in parallel.     

Does your body count? Embodied influences on the preferred counting direction of preschoolers

Many studies confirm that preschoolers from left-to-right reading societies count from left to right, which reflects the cultural direction of their spatial-numerical associations (SNAs). Other factors, not directly related to cultural experience, have not yet been systematically examined as potential determinants of counting direction in preschool children. In this study, we test one such determinant, i.e. whether and how counting direction is related to body-centred reference frames, such as a preferred hand in different spatial-motoric settings. We first asked preschoolers to count spontaneously. Then, we asked them to use a particular hand to count objects presented close to or far away from their bodies. An overall left-to-right counting dominance was replicated, but it was modulated by a counting hand in both tasks: Children preferred to start counting from an object ipsilateral to the hand used. This pattern increased when stimuli were in a position close to the body. We conclude that early counting preferences of preliterate children are embodied. The SNAs which were assessed are not only related to culture but also to situated individual bodily reference frames.     

Physiological threat responses predict number processing

Psychological Research - Being able to adequately process numbers is a key competency in everyday life. Yet, self-reported negative affective responses towards numbers are known to deteriorate...     

Learning of sensory sequences in cerebellar patients

A possible role of the cerebellum in detecting and recognizing event sequences has been proposed. The present study sought to determine whether patients with cerebellar lesions are impaired in the acquisition and discrimination of sequences of sensory stimuli of different modalities. A group of 26 cerebellar patients and 26 controls matched for age, sex, handedness, musicality, and level of education were tested. Auditory and visual sensory sequences were presented out of different sensory pattern categories (tones with different acoustic frequencies and durations, visual stimuli with different spatial locations and colors, sequential vision of irregular shapes) and different ranges of inter-cue time intervals (fast and slow). Motor requirements were small, with vocal responses and no time restrictions. Perception of visual and acoustic stimuli was generally preserved in patients and controls. The number of errors was significantly higher in the faster tempo of sequence presentation in learning of sequences of tones of different frequencies and in learning of sequences of visual stimuli of different spatial locations and different colors. No difference in tempo between the groups was shown. The total number of errors between the two groups was identical in the sequence conditions. No major disturbances in acquisition or discrimination of various sensory sequences were observed in the group of cerebellar patients. Sequence learning may be impaired only in tasks with significant motor demands.     

On the perceptual generality of the unit-decade compatibility effect

Number magnitude is assumed to be holistically represented along a single mental number line. Recently, we have observed a unit-decade-compatibility effect which is inconsistent with that assumption (Nuerk, Weger, & Willmes, 2001). In two-digit Arabic number comparison, we have demonstrated that compatible comparisons in which separate decade and unit comparisons lead to the same decision (32_47, 3 < 4 and 2 < 7) were faster than incompatible trials (37_52, 3 < 5, but 7 > 2). Because overall distance was matched, a holistic model could not account for the compatibility effect. However, one could argue that the compatibility effect was due to the specific vertical perceptual arrangement of the two-digit numbers in Nuerk et al.'s (2001) experiment where the decade digits and unit digits were presented column-wise above each other. To examine this objection, we studied the perceptual generality of the compatibility effect with diagonal presentation. We replicated the compatibility effect with diagonal presentation. It is concluded that the compatibility effect is not due to encoding characteristics imposed by the perceptual setting of the original experiment. In particular, the assumption of an overall analog magnitude representation for two-digit numbers is not consistent with these data.