Xinlin Zhou

Fractions: the new frontier for theories of numerical development

Recent research on fractions has broadened and deepened theories of numerical development. Learning about fractions requires children to recognize that many properties of whole numbers are not true of numbers in general and also to recognize that the one property that unites all real numbers is that they possess magnitudes that can be ordered on number lines. The difficulty of attaining this understanding makes the acquisition of knowledge about fractions an important issue educationally, as well as theoretically. This article examines the neural underpinnings of fraction understanding, developmental and individual differences in that understanding, and interventions that improve the understanding. Accurate representation of fraction magnitudes emerges as crucial both to conceptual understanding of fractions and to fraction arithmetic.

Dissociated brain organization for single-digit addition and multiplication

This study compared the patterns of brain activation elicited by single-digit addition and multiplication problems. 20 Chinese undergraduates were asked to verify whether arithmetic equations were true or false during functional magnetic resonance imaging. Results showed that both addition and multiplication were supported by a broad neural system that involved regions within SMA, precentral gyrus, intraparietal sulcus, occipital gyri, superior temporal gyrus, and middle frontal gyrus, as well as some subcortical structures. Nevertheless, addition problems elicited more activation in the intraparietal sulcus and middle occipital gyri at the right hemisphere, and superior occipital gyri at both hemispheres, whereas multiplication had more activation in precentral gyrus, supplementary motor areas, and posterior and anterior superior temporal gyrus at the left hemisphere. This pattern of dissociated activation supports our hypothesis that addition has greater reliance on visuospatial processing and multiplication on verbal processing.

Gender Differences in Children’s Arithmetic Performance Are Accounted for by Gender Differences in Language Abilities

Studies have shown that female children, on average, consistently outperform male children in arithmetic. In the research reported here, 1,556 pupils (8 to 11 years of age) from urban and rural regions in the greater Beijing area completed 10 cognitive tasks. Results showed that girls outperformed boys in arithmetic tasks (i.e., simple subtraction, complex multiplication), as well as in numerosity-comparison, number-comparison, number-series-completion, choice reaction time, and word-rhyming tasks. Boys outperformed girls in a mental rotation task. Controlling for scores on the word-rhyming task eliminated gender differences in arithmetic, whereas controlling for scores on numerical-processing tasks (number comparison, numerosity estimation, numerosity comparison, and number-series completion) and general cognitive tasks (choice reaction time, Raven’s Progressive Matrices, and mental rotation) did not. These results suggest that girls’ advantage in arithmetic is likely due to their advantage in language processing.

Chinese kindergartners' automatic processing of numerical magnitude in Stroop-like tasks

Using Stroop-like tasks, this study examined whether Chinese kindergartners showed automatic processing of numerical magnitude. A total of 36 children (mean age 5 5 years 10 months) were asked to perform physical size comparison (i.e., “Which of two numbers is bigger in physical size?”) and numerical magnitude tasks (i.e., “Which of two numbers is bigger in numerical magnitude?”) on 216 number pairs. These number pairs varied in levels of congruence between numerical magnitude and physical size (for Stroop effect) and numerical distance (for distance effect). On the basis of analyses of response time and error rates, we found that Chinese kindergartners showed automatic processing of numerical magnitude. These results are significantly different from previous studies’ findings about the onset age (ranging from around the end of first grade to third grade) for automatic processing of numerical magnitude.

Cross-cultural investigation into cognitive underpinnings of individual differences in early arithmetic

The present study evaluated 626 5-7-year-old children in the UK, China, Russia, and Kyrgyzstan on a cognitive test battery measuring: (1) general skills; (2) non-symbolic number sense; (3) symbolic number understanding; (4) simple arithmetic - operating with numbers; and (5) familiarity with numbers. Although most inter-population differences were small, 13% of the variance in arithmetic skills could be explained by the sample, replicating the pattern, previously found with older children in PISA. Furthermore, the same cognitive skills were related to early arithmetic in these diverse populations. Only understanding of symbolic number explained variation in mathematical performance in all samples. We discuss the results in terms of potential influences of socio-demographic, linguistic and genetic factors on individual differences in mathematics.

Development of spatial representation of numbers: A study of the SNARC effect in Chinese children

Using the standard parity judgment task, this study investigated the development of numerical-spatial representation. Participants were 314 healthy right-handed Chinese children (from kindergarteners to sixth graders) and adults. The results revealed that all age groups showed a significant (or marginally significant in the case of first graders) SNARC (spatial-numerical association of response codes) effect, indicating that Chinese children as young as kindergarteners already had developed automatic spatial representations of numbers (or the mental number line). Surprisingly, however, the size of the SNARC effect did not show much developmental change. These results are discussed in the context of the literature on spatial representations of numbers and on cross-cultural differences in early development of number cognition.

Neural substrates for forward and backward recitation of numbers and the alphabet: A close examination of the role of intraparietal sulcus and perisylvian areas

Despite numerous studies on the neural basis of numerical processing, few studies have examined the neural substrates of one of the most basic numerical processing-number sequence recitation. The present study used fMRI to investigate neural substrates of number sequence recitation, focusing on the intraparietal sulcus (IPS) and perisylvian areas. This study used a 2 (number versus alphabet) x 2 (forward versus backward recitation) design. 12 Chinese undergraduates were asked to recite overtly but gently numerical and alphabetical sequences forward and backward. Results showed that, for both numerical and alphabetic sequences, the left IPS was activated when performing backward recitation, but not when performing forward recitation. In terms of perisylvian areas, all four tasks elicited activation in bilateral superior temporal gyrus and inferior frontal gyrus, but forward recitation elicited greater activation in the left posterior superior temporal gyrus than did backward recitation, whereas backward recitation elicited greater activation in the left inferior frontal gyrus than did forward recitation. These results suggest that forward recitation of numbers and the alphabet is typically based on verbal processing of numbers implemented in the perisylvian area, whereas backward recitation would likely require additional neural resources in the IPS.

Dissociation of subtraction and multiplication in the right parietal cortex: Evidence from intraoperative cortical electrostimulation

Previous research has consistently shown that the left parietal cortex is critical for numerical processing, but the role of the right parietal lobe has been much less clear. This study used the intraoperative cortical electrical stimulation approach to investigate neural dissociation in the right parietal cortex for subtraction and multiplication. Results showed that multiplication (as well as picture naming) was not affected by the cortical electrical stimulation on all the targeted sites of the right parietal cortex as well as those of the right temporal cortex. In contrast, stimulation at three right parietal sites (two sites in the right inferior parietal lobule and one in the right angular gyrus) impaired performance on simple subtraction problems. This study provided the first evidence from an intraoperative cortical electrical stimulation study to show the dissociation of arithmetic operations in the right parietal cortex. This dissociation between subtraction and multiplication suggests that the right parietal cortex plays a more significant role in quantity processing (subtraction) than in verbal processing (multiplication) in numerical processing.

Development of fraction concepts and procedures in U.S. and Chinese children

We compared knowledge of fraction concepts and procedures among sixth and eighth graders in China and the United States. As anticipated, Chinese middle school children had higher knowledge of fraction concepts and procedures than U.S. children in the same grades, and the difference in procedural knowledge was much larger than the difference in conceptual knowledge. Of particular interest, national differences in knowledge of fraction concepts were fully mediated by differences in knowledge of fraction procedures, and differences between the knowledge of Chinese and U.S. children were most pronounced among the lowest achieving children within each country. Based on these and previous results, a theoretical model of the mutually facilitative interaction between conceptual and procedural knowledge of fractions is proposed and discussed.

Dissociated neural correlates of quantity processing of quantifiers, numbers, and numerosities

Quantities can be represented using either mathematical language (i.e., numbers) or natural language (i.e., quantifiers). Previous studies have shown that numerical processing elicits greater activation in the brain regions around the intraparietal sulcus (IPS) relative to other semantic processes. However, little research has been conducted to investigate whether the IPS is also critical for the semantic processing of quantifiers in natural language. In this study, 20 adults were scanned with functional magnetic resonance imaging while they performed semantic distance judgment involving six types of materials (i.e., frequency adverbs, quantity pronouns and nouns, animal names, Arabic digits, number words, and dot arrays). Conjunction analyses of brain activation showed that numbers and dot arrays elicited greater activation in the right IPS than did words (i.e., animal names) or quantifiers (i.e., frequency adverbs and quantity pronouns and nouns). Quantifiers elicited more activation in left middle temporal gyrus and inferior frontal gyrus than did numbers and dot arrays. No differences were found between quantifiers and animal names. These findings suggest that, although quantity processing for numbers and dot arrays typically relies on the right IPS region, quantity processing for quantifiers typically relies on brain regions for general semantic processing. Thus, the IPS does not appear to be the only brain region for quantity processing. Hum Brain Mapp 35:444–454, 2014.