Abstract: Number symbols play an very important role in our daily life, we use them to counting, label, rank, and we rely on them to develop superior mathematical skills. But there exits wide variety in the levels of mathematical difference among us. Over the past decades, evidence from cognitive development, comparative cognition, cross-culture cognition and neurobiology has proved numerical skills or number symbols we used in our daily life build on the Approximate Number System(ANS), which represents numbers in an nonverbal and noisy way. It encodes the numerosities of discrete objects or events as analog magnitudes that can be modeled as overlapping Gaussian distributions of activations. Recent years, evidence from the normal people of different ages or children with mathematical learning disability has revealed that the acuity of ANS maybe a more important factor to determine the individuals’ difference in mathematics ability compared with general cognitive ability during our life span. These studies holds that the ANS is instrumental in the acquisition of symbolic numerical skills like arithmetic. People with greater precision in the ANS are more likely to acquire the counting sequence and other subsequent symbolic numerical skills more easily in their childhood, and this may leading to a better symbolic number representations. This article first reviewed the studies about the relationship between the acuity of approximate number system and math ability, including cross-sectional studies, longitudinal studies, training studies and cognitive neuroscience studies. Then we analyze factors that impact this relationship, including age, math ability, inhibitory control. But from these studies we could not conclude when and how ANS representations integrate with math ability. So we then summarize various different hypothesis to explain this relationship. The first hypothesis is that better precision of the ANS representations may lead to increased engagement in number-related activities, which may lead to an decrease in math anxiety and a increase in math ability. The second interpretation of this relationship is that participation in mathematical tasks allows children an opportunity to engage their ANS in a context in which the ANS is likely to be of particular benefit, participating in mathematical activities that engage the ANS may facilitate further development of the ANS and increase the likelihood that children rely on their ANS to learn new mathematical concepts. The third hypothesis holds that the number symbolic number knowledge mediates the relation between ANS acuity and arithmetic competence, these symbolic number knowledge including the mapping of symbolic number, numerical ordering ability and so on. Though many kinds of evidence had been proposed to support these hypothesis, they could not have a conclusion about which one is right. Future research need to investigate not only the relationship between the acuity of approximate number system and different dimensions of mathematics ability on the basis of more reliable and valid paradigm, but also to clarify the theoretical explanation behind this relationship by using other kinds of experiment design like training study. Besides, apply research conclusions to math teaching and the intervention targeting children with mathematical learning disability.

Key words: approximate number system, mathematics ability, mental number line, non-symbolic magnitude comparison task, weber fraction

摘要： 近年来，来自认知发展、比较认知、跨文化认知和神经生物学的研究证据都表明近似数量系统的存在，并且相较于一般认知能力，它更可能是决定个体数学能力差异最为重要的因素。本文综述了有关近似数量系统敏锐度与数学能力相互关系的横断研究、纵向研究、训练研究及认知神经科学的研究成果，分析了影响二者关系的因素，包括个体年龄、数学能力高低、抑制控制等，并总结了多种理论对二者间显著正相关关系的解释。未来研究需要在确定更具信效度的测量范式的基础上探讨近似数量系统与数学能力各维度的关系，以及这种相互关系背后的原因，并将研究结论运用于数学教学及计算障碍个体的干预。

关键词: 近似数量系统, 数学能力, 心理数字线, 数量比较任务, 韦伯系数