Math Fluency during Primary School.

Math fluency is the ability to solve arithmetic facts quickly and accurately (i.e., addition and subtraction problems up to 20, and multiplication and division problems from the multiplication table). Curricula in primary school devote a significant period of time for learning and retrieval of arithmetic facts. Recently, a new computerized tool to assess math fluency-the BGU-MF (Ben-Gurion University Math Fluency) test-was developed and found to be a reliable and valid tool for adults. In the current study, we examine the performance of first to sixth-grade children in math fluency using the BGU-MF. The results present the performance of MF during childhood and emphasize that it continues to develop during primary school. Importantly, proficiency of MF differed by operations, and the automaticity of math facts was acquired in different grades. Moreover, we found that the BGU-MF is a reliable and valid tool not only for adults but also for children during primary school. Our study has educational implications for the teaching, practice, and retrieval of arithmetic facts.

Understanding Estimations of Magnitudes: An fMRI Investigation.

The current study examined whether discrete numerical estimation is based on the same cognitive process as estimation of continuous magnitudes such as weight and time. While the verbal estimation of numerical quantities has a contingent unit of measurement (e.g., how many cookies fit in a cookie jar? _X_ cookies), estimation of time and weight does not (e.g., how much time does it take to fill a bath with water? _X_ minutes/hours/seconds). Therefore, estimation of the latter categories has another level of difficulty, requiring extensive involvement of cognitive control. During a functional magnetic resonance imaging (fMRI) scan, 18 students performed estimations with three estimation categories: number, time, and weight. Estimations elicited activity in multiple brain regions, mainly: (1) visual regions including bilateral lingual gyrus), (2) parietal regions including the left angular gyrus and right supramarginal gyrus, and (3) the frontal regions (cingulate gyrus and the inferior frontal cortex). Continuous magnitude estimations (mostly time) produced different frontal activity than discrete numerical estimations did, demonstrating different profiles of brain activations between discrete numerical estimations and estimations of continuous magnitudes. The activity level in the right middle and inferior frontal gyrus correlated with the tendency to give extreme responses, signifying the importance of the right prefrontal lobe in estimations.

Damage to the Intraparietal Sulcus Impairs Magnitude Representations of Results of Complex Arithmetic Problems.

Past research investigating the role of the intraparietal sulcus (IPS) in numerical processes focused mainly on quantity and numerical comparisons as well on single digit arithmetic. The present study investigates the involvement of the IPS in estimating the results of multi-digit multiplication problems. For this purpose, the performance a 24-year-old female (JD) with brain damage in the left IPS was compared to an age-matched control group in the computation estimation task. When required to estimate whether the results of multi-digit multiplication problems are smaller or larger than given reference numbers, JD, in contrast to controls, did not show the common patterns of distance and size effects. Her strategy use was also atypical. Most control participants used both the approximated calculation strategy that involves rounding and calculation procedures and the sense of magnitude strategy that relies on an intuitive approximated magnitude representation of the results. In contrast, JD used only the former but not the latter strategy. Together, these findings suggest that the damage to the IPS impaired JD's representations of magnitude that play an important role in this computation estimation task.

Modal-independent Pattern Recognition Deficit in Developmental Dyscalculia Adults: Evidence from Tactile and Visual Enumeration.

Developmental dyscalculia (DD) is characterized by lower numerical and finger-related skills. Studies of enumeration among those DD that suggested core deficiency in pattern recognition, working memory or/and attention were mostly carried out in the visual modality. In our study, we examined visual (dots) enumeration of 1-10 stimuli and tactile (vibration) enumeration of 1-10 fingers among DD and matched-control adults. We used 800-ms stimuli exposure time of either random/non-neighboring or canonical/neighboring stimuli arrangements (visual/tactile). Compared to controls, those with DD responded faster in visual random enumeration and did not differ in reaction time (RT) of canonical stimuli arrangements. However, while the control group had near perfect accuracy in random stimuli arrangements of up to five stimuli, DD participants performed accurately for only up to four stimuli, and they were less accurate in the canonical stimuli arrangements in the counting range. In the tactile task, DD participants showed less accurate tactile enumeration only for neighboring arrangements, more profoundly for finger counting (FC) patterns. The longer exposure time in the visual task enabled us to explore pattern recognition effects when working memory and attention loads were low. We discuss possible modal-independent deficits in pattern recognition and working memory on enumeration performance among those with DD and the unique role of fingers in ordinal and cardinal representation of numbers.

Enumeration and Alertness in Developmental Dyscalculia.

Enumeration, the ability to report an amount of elements, differs as a function of range. Subitizing (quantities 1-4) is an accurate and quick process with reaction times (RTs) minimally affected by the number of presented elements within its range. In the counting range (range of 5-9 elements), RTs increase linearly. Subitizing was considered to be a pre-attentive process for many years. However, recently we found that subitizing could be facilitated by improving engagement of attention. Specifically, brief alerting cues increase attentional engagement and reduced RTs in the subitizing range. Moreover, previous studies found that students with developmental dyscalculia (DD) have a smaller than normal subitizing range (3 vs. 4) and their alerting attentional system is impaired. In the current study, we explored whether an alerting cue would increase the subitizing range of adults suffering from DD from 3 to 4. For controls, alerting increased accuracy rates and facilitated enumeration of quantities only in the subitizing range. Participants with DD presented a larger alerting effect; an alerting cue enhanced their RTs in all ranges, but did not increase their smaller than normal subitizing range or accuracy. Our results suggest that both domain-general and domain-specific abilities contribute to the mechanism of enumeration and related to developmental dyscalculia.

Tactile enumeration: A case study of acalculia.

Enumeration is one of the building blocks of arithmetic and fingers are used as a counting tool in early steps. Subitizing-fast and accurate enumeration of small quantities-has been vastly studied in the visual modality, but less in the tactile modality. We explored tactile enumeration using fingers, and gray matter (GM) changes using voxel-based morphometry (VBM), in acalculia. We examined JD, a 22-year-old female with acalculia following a stroke to the left inferior parietal cortex. JD and a neurologically healthy normal comparison (NC) group reported how many fingers were stimulated. JD was tested at several time points, including at acute and chronic phases. Using the sensory intact hand for tactile enumeration, JD showed deficit in the acute phase, compared to the NC group, and improvement in the chronic phase of (1) the RT slope of enumerating up to four stimuli, (2) enumerating neighboring fingers, and (3) arithmetic fluency performance. Moreover, VBM analysis showed a larger GM volume for JD relative to the NC group in the right middle occipital cortex, most profoundly in the chronic phase. JD's performance serves as a first glance of tactile enumeration in acalculia. Pattern-recognition-based results support the suggestion of subitizing being the enumeration process when using one hand. Moreover, the increase in GM in the occipital cortex lays the groundwork for studying the innate and primitive ability to perceive and evaluate sizes or amounts-"sense of magnitude"- as a multisensory magnitude area and as part of a recovery path for deficits in basic numerical abilities.

Conceptual size in developmental dyscalculia and dyslexia.

OBJECTIVE: People suffering from developmental dyscalculia (DD) are known to have impairment in numerical abilities and have been found to have weaker processing of countable magnitudes. However, not much research was done on their abilities to process noncountable magnitudes. An example of noncountable magnitude is conceptual size (e.g., mouse is small and elephant is big). Recently, we found that adults process conceptual size automatically. The current study examined automatic processing of conceptual size in students with DD and developmental dyslexia. METHOD: Conceptual and physical sizes were manipulated orthogonally to create congruent (e.g., a physically small apple compared to a physically large violin) and incongruent (e.g., a physically large apple compared to a physically small violin) conditions. Participants were presented with 2 objects and had to choose the larger one. Each trial began with an instruction to respond to the physical or to the conceptual dimension. RESULTS: Control and the dyslexic groups presented automatic processing of both conceptual and physical sizes. The dyscalculic group presented automatic processing of physical size but not automaticity of processing conceptual size. CONCLUSION: Our results fit with previous findings of weaker magnitude representation in those with DD, specifically regarding noncountable magnitudes, and support theories of a shared neurocognitive substrate for different types of magnitudes. (PsycINFO Database Record

Visual and Imagery Magnitude Comparisons Are Affected Following Left Parietal Lesion.

We describe Jane Dow (JD), a young right-handed female with acalculia following a cerebral infarction in the left intraparietal sulcus. We investigated automatic processing of different types of magnitudes that were presented visually or through imagery. We employed the size congruity task and the mental clock task that differ in stimuli presentation and in working memory load. In the size congruity task, for physical comparisons, JD presented a lack of facilitation effect, suggesting a deficit in the automatic processing of numerical values. In the mental clock task, JD performed as accurate as controls did but much slower. In both tasks, JD presented a steeper distance effect compared to controls, suggesting a deficit in a domain-general comparison process. Our findings present an atypical pattern of magnitude processing following a left parietal lesion that appears not only for visually presented stimuli but also for imagery-based magnitudes. These finding support recent theories suggesting different types of magnitudes are interconnected with each other.

Alerting cues enhance the subitizing process.

Enumeration of elements differs as a function of their range. Subitizing (quantities 1-4) is considered to be an accurate and quick process with reaction times minimally affected by the number of presented elements within its range. In contrast, small estimation (range of 5-9 elements exposed briefly) is a less precise linear process. Subitizing was consider to be a pre-attentive process for many years. However, recent studies found that when attentional resources were occupied elsewhere, the subitizing process was impaired. In the current study, we examined whether subitizing can be facilitated by improving engagement of attention. Specifically, brief alerting cues that increase attentional engagement were presented in half of the trials during enumeration tasks. In Experiment 1, participants were required to enumerate dots presented in random arrays within the subitizing or small estimation range. Alerting facilitated enumeration of quantities in the subitizing range, but not in the small estimation range. We suggested that the benefit of alerting on the subitizing process was achieved via enhancement of global processing, a process that was previously associated with both alerting and subitizing. In Experiment 2, we provided direct evidence for this hypothesis by demonstrating that when global processing was used for items in the small estimation range (i.e., presenting quantities in a canonical array), a subitizing-like pattern was revealed in quantities beyond the subitizing range.

Automaticity of Conceptual Magnitude.

What is bigger, an elephant or a mouse? This question can be answered without seeing the two animals, since these objects elicit conceptual magnitude. How is an object's conceptual magnitude processed? It was suggested that conceptual magnitude is automatically processed; namely, irrelevant conceptual magnitude can affect performance when comparing physical magnitudes. The current study further examined this question and aimed to expand the understanding of automaticity of conceptual magnitude. Two different objects were presented and participants were asked to decide which object was larger on the screen (physical magnitude) or in the real world (conceptual magnitude), in separate blocks. By creating congruent (the conceptually larger object was physically larger) and incongruent (the conceptually larger object was physically smaller) pairs of stimuli it was possible to examine the automatic processing of each magnitude. A significant congruity effect was found for both magnitudes. Furthermore, quartile analysis revealed that the congruity was affected similarly by processing time for both magnitudes. These results suggest that the processing of conceptual and physical magnitudes is automatic to the same extent. The results support recent theories suggested that different types of magnitude processing and representation share the same core system.