非符号数量表征和符号分数表征的关系

doi:10.3724/SP.J.1042.2021.02161

摘要/Abstract

摘要：

个体学习符号分数的一个关键是能对其数值形成准确表征。现有研究假设符号分数表征的认知基础是人类自婴幼儿期就具有的非符号数量表征(如表征两个集合各自的数量, 或两个数量的比例)。其证据包括表征非符号数量(尤其是非符号数量比例关系)和表征符号分数在行为和大脑神经活动层面上都表现出相关性。然而要说明非符号数量表征是符号分数表征的认知基础, 还需更多研究表明两者在数量概念上的独特相关和因果联系, 并阐明符号分数表征形成的认知机制。

关键词: 非符号数量表征, 比例加工系统, 非符号比例表征, 非符号分数表征, 符号分数表征

Abstract:

Representing the numerical value of symbolic fraction is central to the conceptual learning of symbolic fraction. The representation of symbolic fraction has been hypothesized to be built on non- symbolic magnitude representation (e.g., representing the magnitude of a set, or the proportion between two magnitudes) that exists since human early infancy. The supportive evidence includes that non-symbolic magnitude representation, especially the non-symbolic representation for the proportion between quantities, is correlated with the representation for symbolic fraction in behavioral and brain neural responses. However, further research is needed to determine the unique and causal relation between non-symbolic magnitude representation and symbolic fraction representation, and to clarify the specific cognitive mechanism underlying symbolic fraction representation.

Key words: non-symbolic magnitude representation, ratio processing system, non-symbolic proportional representation, non-symbolic fraction representation, symbolic fraction representation

中图分类号:

B844
B842

毛伙敏, 刘琴, 吕建相, 牟毅. (2021). 非符号数量表征和符号分数表征的关系. 心理科学进展 , 29(12), 2161-2171.

MAO Huomin, LIU Qin, LÜ Jianxiang, MOU Yi. (2021). The relation between non-symbolic magnitude representation and symbolic fraction representation. Advances in Psychological Science, 29(12), 2161-2171.

参考文献 92

[1]	高瑞彦, 牛美心, 杨涛, 周新林. (2018). 4-8年级学生分数数量表征的准确性及形式. 心理发展与教育, 34(4), 443-452. https://doi.org/10.16187/j.cnki.issn1001-4918.2018.04.08
[2]	高婷, 刘儒德, 刘颖, 庄鸿娟. (2016). 小学生分数比较中的加工模式: 基于反应时和口语报告的研究. 心理发展与教育, 32(4), 463-470. https://doi.org/10.16187/j.cnki.issn1001-4918.2016.04.10
[3]	刘春晖, 辛自强. (2010). 分数认知的“整数偏向”研究: 理论与方法. 心理科学进展, 18(1), 65-74.
[4]	孙玉, 司继伟, 黄碧娟. (2016). 分数的数量表征. 心理科学进展, 24(4), 1207-1216. https://doi.org/10.3724/SP.J.1042.2016.01207
[5]	辛自强, 韩玉蕾. (2014). 小学低年级儿童的等值分数概念发展及干预. 心理学报, 46(6), 791-806. https://doi.org/10.3724/SP.J.1041.2014.00791
[6]	辛自强, 刘国芳. (2011). 非符号分数与整数计算能力的发展及其与数字记忆的关系. 心理科学, 34(3), 520-526. https://doi.org/CNKI:SUN:XLKX.0.2011-03-003
[7]	杨伊生, 刘儒德. (2008). 儿童分数概念发展研究综述. 内蒙古师范大学学报(教育科学版), 21(6), 130-134.
[8]	张丽, 卢彩芳, 杨新荣. (2014). 3-6年级儿童整数数量表征与分数数量表征的关系. 心理发展与教育, 30(1), 1-8. https://doi.org/CNKI:SUN:XLFZ.0.2014-01-001
[9]	张丽, 辛自强, 王琦, 李红. (2012). 整数构成对分数加工的影响. 心理发展与教育, 28(1), 31-38. https://doi.org/CNKI:SUN:XLFZ.0.2012-01-005
[10]	Bailey, D. H., Hoard, M. K., Nugent, L., & Geary, D. C.(2012). Competence with fractions predicts gains in mathematics achievement. Journal of Experimental Child Psychology, 113(3), 447-455. https://doi.org/10.1016/j.jecp.2012.06.004 doi: 10.1016/j.jecp.2012.06.004 URL pmid: 22832199
[11]	Bailey, D. H., Siegler, R. S., & Geary, D. C.(2014). Early predictors of middle school fraction knowledge. Developmental Science, 17(5), 775-785. https://doi.org/10.1111/desc.12155 doi: 10.1111/desc.12155 URL pmid: 24576209
[12]	Barner, D.(2017). Language, procedures, and the non- perceptual origin of number word meanings. Journal of Child Language, 44(3), 553-590. https://doi.org/10.1017/S0305000917000058 doi: 10.1017/S0305000917000058 URL pmid: 28376934
[13]	Begolli, K. N., Booth, J. L., Holmes, C. A., & Newcombe, N. S.(2020). How many apples make a quarter? The challenge of discrete proportional formats. Journal of Experimental Child Psychology, 192, 104774. https://doi.org/10.1016/j.jecp.2019.104774 doi: 10.1016/j.jecp.2019.104774 URL
[14]	Bhatia, P., Delem, M., Léone, J., Boisin, E., Cheylus, A., Gardes, M.-L., & Prado, J.(2020). The ratio processing system and its role in fraction understanding: Evidence from a match-to-sample task in children and adults with and without dyscalculia. Quarterly Journal of Experimental Psychology, 73(12), 2158-2176. https://doi.org/10.1177/1747021820940631 doi: 10.1177/1747021820940631 URL
[15]	Boyer, T. W., & Levine, S. C.(2012). Child proportional scaling: Is 1/3=2/6=3/9=4/12?. Journal of Experimental Child Psychology, 111(3), 516-533. https://doi.org/10.1016/j.jecp.2011.11.001 doi: 10.1016/j.jecp.2011.11.001 URL pmid: 22154533
[16]	Boyer, T. W., & Levine, S. C.(2015). Prompting children to reason proportionally: Processing discrete units as continuous amounts. Developmental Psychology, 51(5), 615-620. https://doi.org/10.1037/a0039010 doi: 10.1037/a0039010 URL pmid: 25751097
[17]	Boyer, T. W., Levine, S. C., & Huttenlocher, J.(2008). Development of proportional reasoning: Where young children go wrong. Developmental Psychology, 44(5), 1478-1490. https://doi.org/10.1037/a0013110 doi: 10.1037/a0013110 URL pmid: 18793078
[18]	Braithwaite, D. W., & Siegler, R. S.(2018). Children learn spurious associations in their math textbooks: Examples from fraction arithmetic. Journal of Experimental Psychology: Learning Memory and Cognition, 44(11), 1765-1777. https://doi.org/10.1037/xlm0000546 doi: 10.1037/xlm0000546 URL
[19]	Braithwaite, D. W., Tian, J., & Siegler, R. S.(2018). Do children understand fraction addition?. Developmental Science, 21(4), e12601. https://doi.org/10.1111/desc.12601 doi: 10.1111/desc.2018.21.issue-4 URL
[20]	Chen, Q. X., & Li, J. G.(2014). Association between individual differences in non-symbolic number acuity and math performance: A meta-analysis. Acta Psychologica, 148, 163-172. https://doi.org/10.1016/j.actpsy.2014.01.016 doi: 10.1016/j.actpsy.2014.01.016 URL
[21]	Chu, F. W., vanMarle, K., & Geary, D. C.(2016). Predicting children's reading and mathematics achievement from early quantitative knowledge and domain-general cognitive abilities. Frontiers in Psychology, 7, 775. https://doi.org/10.3389/fpsyg.2016.00775
[22]	Cui, J. X., Li, L. N., Li, M. Y., Siegler, R., & Zhou, X. L.(2020). Middle temporal cortex is involved in processing fractions. Neuroscience Letters, 725, 134901. https://doi.org/10.1016/j.neulet.2020.134901 doi: 10.1016/j.neulet.2020.134901 URL
[23]	Dehaene, S., Piazza, M., Pinel, P., & Cohen, L.(2003). Three parietal circuits for number processing. Cognitive Neuropsychology, 20(3), 487-506. https://doi.org/10.1080/02643290244000239 doi: 10.1080/02643290244000239 URL
[24]	Denison, S., Reed, C., & Xu, F.(2013). The emergence of probabilistic reasoning in very young infants: Evidence from 4.5- and 6-month-olds. Developmental Psychology, 49(2), 243-249. https://doi.org/10.1037/a0028278 doi: 10.1037/a0028278 URL
[25]	Denison, S., & Xu, F.(2014). The origins of probabilistic inference in human infants. Cognition, 130(3), 335-347. https://doi.org/10.1016/j.cognition.2013.12.001 doi: 10.1016/j.cognition.2013.12.001 URL
[26]	de Smedt, B., Noël, M.-P., Gilmore, C., & Ansari, D.(2013). How do symbolic and non-symbolic numerical magnitude processing skills relate to individual differences in children's mathematical skills? A review of evidence from brain and behavior. Trends in Neuroscience and Education, 2(2), 48-55. doi: 10.1016/j.tine.2013.06.001 URL
[27]	DeWolf, M., Bassok, M., & Holyoak, K. J.(2015). Conceptual structure and the procedural affordances of rational numbers: Relational reasoning with fractions and decimals. Journal of Experimental Psychology: General, 144(1), 127-150. https://doi.org/10.1037/xge0000034 doi: 10.1037/xge0000034 URL
[28]	Drucker, C. B., Rossa, M. A., & Brannon, E. M.(2016). Comparison of discrete ratios by rhesus macaques (Macaca mulatta). Animal Cognition, 19(1), 75-89. https://doi.org/10.1007/s10071-015-0914-9 doi: 10.1007/s10071-015-0914-9 URL
[29]	Elliott, L., Feigenson, L., Halberda, J., & Libertus, M. E.(2018). Bidirectional, longitudinal associations between math ability and approximate number system precision in childhood. Journal of Cognition and Development, 20(1), 56-74. https://doi.org/10.1080/15248372.2018.1551218 doi: 10.1080/15248372.2018.1551218 URL
[30]	Fazio, L. K., Bailey, D. H., Thompson, C. A., & Siegler, R. S.(2014). Relations of different types of numerical magnitude representations to each other and to mathematics achievement. Journal of Experimental Child Psychology, 123, 53-72. https://doi.org/10.1016/j.jecp.2014.01.013 doi: 10.1016/j.jecp.2014.01.013 URL pmid: 24699178
[31]	Feigenson, L., Dehaene, S., & Spelke, E.(2004). Core systems of number. Trends in Cognitive Sciences, 8(7), 307-314. https://doi.org/10.1016/j.tics.2004.05.002 URL pmid: 15242690
[32]	Fuhs, M. W., & McNeil, N. M.(2013). ANS acuity and mathematics ability in preschoolers from low-income homes: Contributions of inhibitory control. Developmental Science, 16(1), 136-148. https://doi.org/10.1111/desc.12013 doi: 10.1111/desc.2012.16.issue-1 URL
[33]	Gallistel, C. R.(2007). Commentary on Le Corre & Carey. Cognition, 105(2), 439-445. https://doi.org/10.1016/j.cognition.2007.01.010 doi: 10.1016/j.cognition.2007.01.010 URL
[34]	Gallistel, C. R., & Gelman, I. I.(2000). Non-verbal numerical cognition: From reals to integers. Trends in Cognitive Sciences, 4(2), 59-65. https://doi.org/10.1016/s1364-6613(99)01424-2 URL pmid: 10652523
[35]	Geary, D. C., & vanMarle, K.(2016). Young children's core symbolic and nonsymbolic quantitative knowledge in the prediction of later mathematics achievement. Developmental Psychology, 52(12), 2130-2144. https://doi.org/10.1037/dev0000214 doi: 10.1037/dev0000214 URL
[36]	Gilmore, C., Attridge, N., Clayton, S., Cragg, L., Johnson, S., Marlow, N., … Inglis, M.(2013). Individual differences in inhibitory control, not non-verbal number acuity, correlate with mathematics achievement. PLOS ONE, 8(6), e67374. https://doi.org/10.1371/journal.pone.0067374 doi: 10.1371/journal.pone.0067374 URL
[37]	Gilmore, C., & Cragg, L.(2018). Heterogeneity of function in numerical cognition. . In H. Avishai, & F. Wim (Eds.), The role of executive function skills in the development of children's mathematical competencies (pp. 263-286)Elsevier. https://doi.org/10.1016/B978-0-12-811529-9.00014-5
[38]	Goswami, U.(1989). Relational complexity and the development of analogical reasoning. Cognitive Development, 4(3), 251-268. https://doi.org/10.1016/0885-2014(89)90008-7
[39]	Gouet, C., Carvajal, S., Halberda, J., & Peña, M.(2020). Training nonsymbolic proportional reasoning in children and its effects on their symbolic math abilities. Cognition, 197, 104154. https://doi.org/10.1016/j.cognition.2019.104154 doi: 10.1016/j.cognition.2019.104154 URL
[40]	Hansen, N., Jordan, N. C., Fernandez, E., Siegler, R. S., Fuchs, L., Gersten, R., & Micklos, D.(2015). General and math-specific predictors of sixth-graders' knowledge of fractions. Cognitive Development, 35, 34-49. https://doi.org/10.1016/j.cogdev.2015.02.001 doi: 10.1016/j.cogdev.2015.02.001 URL
[41]	Howard, S. R., Avarguès-Weber, A., Garcia, J. E., Greentree, A. D., & Dyer, A. G.(2019). Surpassing the subitizing threshold: Appetitive-aversive conditioning improves discrimination of numerosities in honeybees. The Journal of Experimental Biology, 222(Pt 19), jeb205658. https://doi.org/10.1242/jeb.205658 doi: 10.1242/jeb.205658 URL
[42]	Hyde, D. C., Boas, D. A., Blair, C., & Carey, S.(2010). Near-infrared spectroscopy shows right parietal specialization for number in pre-verbal infants. NeuroImage, 53(2), 647- 652. https://doi:10.1016/j.neuroimage.2010.06.030 doi: https://doi:10.1016/j.neuroimage.2010.06.030 URL
[43]	Hyde, D. C., Khanum, S., & Spelke, E. S.(2014). Brief non- symbolic, approximate number practice enhances subsequent exact symbolic arithmetic in children. Cognition, 131(1), 92-107. https://doi.org/10.1016/j.cognition.2013.12.007 doi: 10.1016/j.cognition.2013.12.007 URL
[44]	Hyde, D. C., & Mou, Y.(2015). Neural and behavioral signatures core numerical abilities and early numerical development. In D. B. Berch, D. C. Geary, & K. Mann Koepke (Eds.), Mathematical cognition and learning, (Vol.2, pp. 51-77)Elsevier. https://doi.org/10.1016/B978-0-12-801871-2.00003-4
[45]	Hyde, D. C., Simon, C. E., Berteletti, I., & Mou, Y.(2016). The relationship between non-verbal systems of number and counting development: A neural signatures approach. Developmental Science, 20(6), https://doi.org/10.1111/desc.12464
[46]	Hyde, D. C., & Spelke, E. S.(2011). Neural signatures of number processing in human infants: Evidence for two core systems underlying numerical cognition. Developmental Science, 14(2), 360-371. https://doi.org/10.1111/j.1467-7687.2010.00987. doi: 10.1111/desc.2011.14.issue-2 URL
[47]	Jacob, S. N., & Nieder, A.(2009a). Notation-independent representation of fractions in the human parietal cortex. The Journal of Neuroscience, 29(14), 4652-4657. https://doi.org/10.1523/JNEUROSCI.0651-09.2009 doi: 10.1523/JNEUROSCI.0651-09.2009 URL
[48]	Jacob, S. N., & Nieder, A.(2009b). Tuning to non-symbolic proportions in the human frontoparietal cortex. European Journal of Neuroscience, 30(7), 1432-1442. https://doi.org/10.1111/j.1460-9568.2009.06932. doi: 10.1111/ejn.2009.30.issue-7 URL
[49]	Jordan, N. C., Hansen, N., Fuchs, L. S., Siegler, R. S., Gersten, R., & Micklos, D.(2013). Developmental predictors of fraction concepts and procedures. Journal of Experimental Child Psychology, 116(1), 45-58. https://doi.org/10.1016/j.jecp.2013.02.001 doi: 10.1016/j.jecp.2013.02.001 URL pmid: 23506808
[50]	Jordan, N. C., Resnick, I., Rodrigues, J., Hansen, N., & Dyson, N.(2017). Delaware longitudinal study of fraction learning: implications for helping children with mathematics difficulties. Journal of Learning Disabilities, 50(6), 621- 630. https://doi.org/10.1177/0022219416662033 doi: 10.1177/0022219416662033 URL pmid: 27506551
[51]	Jordan, N. C., Rodrigues, J., Hansen, N., & Resnick, I.(2017). Fraction development in children: Importance of building numerical magnitude understanding. In D. C. Geary, D. B. Berch, R. J. Ochsendorf, K. M. Koepke, In Mathematical Cognition and Learning, Acquisition of Complex Arithmetic Skills and Higher-Order Mathematics Concepts (pp.125- 140). Elsevier. https://doi.org/10.1016/B978-0-12-801871-2.00003-4.
[52]	Kalra, P. B., Binzak, J. V., Matthews, P. G., & Hubbard, E. M.(2020). Symbolic fractions elicit an analog magnitude representation in school-age children. Journal of Experimental Child Psychology, 195, 104844. https://doi.org/10.1016/j.jecp.2020.104844 doi: 10.1016/j.jecp.2020.104844 URL
[53]	Keller, L., & Libertus, M.(2015). Inhibitory control may not explain the link between approximation and math abilities in kindergarteners from middle class families. Frontiers in Psychology, 6, 685. https://doi.org/10.3389/fpsyg.2015.00685
[54]	LeFevre, J.-A., Fast, L., Skwarchuk, S.-L., Smith-Chant, B. L., Bisanz, J., Kamawar, D., & Penner-Wilger, M.(2010). Pathways to mathematics: Longitudinal predictors of performance. Child Development, 81(6), 1753-1767. https://doi.org/10.1111/j.1467-8624.2010.01508. doi: 10.1111/cdev.2010.81.issue-6 URL
[55]	Leibovich, T., & Ansari, D.(2016). The symbol-grounding problem in numerical cognition: A review of theory, evidence, and outstanding questions. Canadian Journal of Experimental Psychology/Revue Canadienne de Psychologie Expérimentale, 70(1), 12-23. https://doi.org/10.1037/cep0000070 doi: 10.1037/cep0000070 URL
[56]	Libertus, M. E., Odic, D., Feigenson, L., & Halberda, J.(2016). The precision of mapping between number words and the approximate number system predicts children's formal math abilities. Journal of Experimental Child Psychology, 150, 207-226. https://doi.org/10.1016/j.jecp.2016.06.003 doi: S0022-0965(16)30052-2 URL pmid: 27348475
[57]	Lyons, I. M., Bugden, S., Zheng, S., de Jesus, S.,& Ansari, D. (2018). Symbolic number skills predict growth in nonsymbolic number skills in kindergarteners. Developmental Psychology, 54(3), 440-457. https://doi.org/10.1037/dev0000445 doi: 10.1037/dev0000445 URL pmid: 29154653
[58]	Matejko, A. A., & Ansari, D.(2016). Trajectories of symbolic and nonsymbolic magnitude processing in the first year of formal schooling. PlOS ONE, 11(3), e0149863. https://doi.org/10.1371/journal.pone.0149863 doi: 10.1371/journal.pone.0149863 URL
[59]	Matthews, P. G., & Chesney, D. L.(2015). Fractions as percepts? Exploring cross-format distance effects for fractional magnitudes. Cognitive Psychology, 78, 28-56. https://doi.org/10.1016/j.cogpsych.2015.01.006 doi: 10.1016/j.cogpsych.2015.01.006 URL
[60]	Matthews, P. G., Lewis, M. R., & Hubbard, E. M.(2016). Individual differences in nonsymbolic ratio processing predict symbolic math performance. Psychological Science, 27(2), 191-202. https://doi.org/10.1177/0956797615617799 doi: 10.1177/0956797615617799 URL
[61]	McCrink, K., & Wynn, K.(2007). Ratio abstraction by 6- month-old infants. Psychological Science, 18(8), 740-745. https://doi.org/10.1111/j.1467-9280.2007.01969. URL pmid: 17680947
[62]	McCrink, K., & Wynn, K.(2009). Operational momentum in large-number addition and subtraction by 9-month-olds. Journal of Experimental Child Psychology, 103(4), 400- 408. https://doi.org/10.1016/j.jecp.2009.01.013 doi: 10.1016/j.jecp.2009.01.013 URL pmid: 19285683
[63]	Mock, J., Huber, S., Bloechle, J., Bahnmueller, J., Moeller, K., & Klein, E.(2019). Processing symbolic and non- symbolic proportions: Domain-specific numerical and domain-general processes in intraparietal cortex. Brain Research, 1714, 133-146. https://doi.org/10.1016/j.brainres.2019.02.029 doi: 10.1016/j.brainres.2019.02.029 URL
[64]	Mock, J., Huber, S., Bloechle, J., Dietrich, J. F., Bahnmueller, J., Rennig, J., ... Moeller, K.(2018). Magnitude processing of symbolic and non-symbolic proportions: An fMRI study. Behavioral and Brain Functions, 14(1), 9. https://doi.org/10.1186/s12993-018-0141-z doi: 10.1186/s12993-018-0141-z URL
[65]	Mou, Y., Berteletti, I., & Hyde, D. C.(2018). What counts in preschool number knowledge? A Bayes factor analytic approach toward theoretical model development. Journal of Experimental Child Psychology, 166, 116-133. https://doi.org/10.1016/j.jecp.2017.07.016 doi: S0022-0965(16)30265-X URL pmid: 28888192
[66]	Mou, Y., Li, Y. R., Hoard, M. K., Nugent, L. D., Chu, F. W., Rouder, J. N., & Geary, D. C.(2016). Developmental foundations of children's fraction magnitude knowledge. Cognitive Development, 39, 141-153. https://doi.org/10.1016/j.cogdev.2016.05.002 URL pmid: 27773965
[67]	Möhring, W., Newcombe, N. S., Levine, S. C., & Frick, A.(2016). Spatial proportional reasoning is associated with formal knowledge about fractions. Journal of Cognition and Development, 17(1), 67-84. https://doi: 10.1080/15248372.2014.996289 doi: 10.1080/15248372.2014.996289 URL
[68]	Mundy, E., & Gilmore, C. K.(2009). Children's mapping between symbolic and nonsymbolic representations of number. Journal of Experimental Child Psychology, 103(4), 490-502. https://doi.org/10.1016/j.jecp.2009.02.003 doi: 10.1016/j.jecp.2009.02.003 URL
[69]	Negen, J., & Sarnecka, B. W.(2015). Is there really a link between exact-number knowledge and approximate number system acuity in young children?. The British Journal of Developmental Psychology, 33(1), 92-105. https://doi.org/10.1111/bjdp.12071 doi: 10.1111/bjdp.2015.33.issue-1 URL
[70]	Nieder, A.(2016). The neuronal code for number. Nature Reviews Neuroscience, 17(6), 366-382. https://doi.org/10.1038/nrn.2016.40 doi: 10.1038/nrn.2016.40 URL
[71]	Ni, Y. J., & Zhou, Y.-D.(2005). Teaching and learning fraction and rational numbers: The origins and implications of whole number bias. Educational Psychologist, 40(1), 27-52. https://doi.org/10.1207/s15326985ep4001_3 doi: 10.1207/s15326985ep4001_3 URL
[72]	Park, Y., Binzak, J., Toomarian, E., Kalra, P., Matthews, P. G., & Hubbard, E.(2018, July) Developmental changes in children's processing of nonsymbolic ratio magnitudes: A cross-sectional fMRI study. Paper presented to 40th Annual Meetings of Cognitive Science Society, Madison, WI, USA.
[73]	Park, Y., Viegut, A. A., & Matthews, P. G.(2020). More than the sum of its parts: Exploring the development of ratio magnitude versus simple magnitude perception. Developmental Science, e13043. https://doi.org/10.1111/ DESC.13043
[74]	Piazza, M.(2010). Neurocognitive start-up tools for symbolic number representations. Trends in cognitive Sciences, 14(12), 542-551. https://doi.org/10.1016/j.tics.2010.09.008 doi: 10.1016/j.tics.2010.09.008 URL
[75]	Sasanguie, D., Defever, E., Maertens, B., & Reynvoet, B.(2014). The approximate number system is not predictive for symbolic number processing in kindergarteners. Quarterly Journal of Experimental Psychology, 67(2), 271-280. https://doi.org/10.1080/17470218.2013.803581 doi: 10.1080/17470218.2013.803581 URL
[76]	Sasanguie, D., de Smedt, B., Defever, E., & Reynvoet, B..(2012). Association between basic numerical abilities and mathematics achievement. The British Journal of Developmental Psychology, 30(Pt 2), 344-357. https://doi.org/10.1111/j.2044-835X.2011.02048. doi: 10.1111/bjdp.2012.30.issue-2 URL
[77]	Schneider, M., Beeres, K., Coban, L., Merz, S., Schmidt, S. S., Stricker, J., & de Smedt, B.(2017). Associations of non- symbolic and symbolic numerical magnitude processing with mathematical competence: A meta-analysis. Developmental Science, 20(3), https://doi.org/10.1111/desc.12372
[78]	Siegler, R. S., Fazio, L. K., Bailey, D. H., & Zhou, X. L.(2013). Fractions: The new frontier for theories of numerical development. Trends in Cognitive Sciences, 17(1), 13-19. https://doi.org/10.1016/j.tics.2012.11.004 doi: 10.1016/j.tics.2012.11.004 URL pmid: 23219805
[79]	Siegler, R. S., & Pyke, A. A.(2013). Developmental and individual differences in understanding of fractions. Developmental Psychology, 49(10), 1994-2004. https://doi.org/10.1037/a0031200 doi: 10.1037/a0031200 URL pmid: 23244401
[80]	Siegler, R. S., Thompson, C. A., & Schneider, M.(2011). An integrated theory of whole number and fractions development. Cognitive Psychology, 62(4), 273-296. https://doi.org/10.1016/j.cogpsych.2011.03.001 doi: 10.1016/j.cogpsych.2011.03.001 URL pmid: 21569877
[81]	Spelke, E. S.(2011). Natural number and natural geometry. In S. Dehaene, & E. M. Brannon (Eds.), Space, time and number in the brain (pp. 287-317)Elsevier. https://doi.org/10.1016/B978-0-12-385948-8.00018-9
[82]	Spelke, E. S.(2017). Core knowledge, language, and number. Language Learning and Development, 13(2), 147-170. https://doi.org/10.1080/15475441.2016.1263572 doi: 10.1080/15475441.2016.1263572 URL
[83]	Starr, A., & Brannon, E. M.(2015). Evolutionary and developmental continuities in numerical cognition. In D. C. Geary, D. B. Berch, & K. M. Koepke (Eds.), Mathematical Cognition and Learning (Vol.1, pp. 123-144). Elsevier. https://doi.org/10.1016/B978-0-12-420133-0.00005-3
[84]	Starr, A., Tomlinson, R. C., & Brannon, E. M.(2018). The acuity and manipulability of the ANS have separable influences on preschoolers' symbolic math achievement. Frontiers in Psychology, 9, 2554. https://doi.org/10.3389/fpsyg.2018.02554 doi: 10.3389/fpsyg.2018.02554 URL
[85]	Suárez-Pellicioni, M., & Booth, J. R.(2018). Fluency in symbolic arithmetic refines the approximate number system in parietal cortex. Human Brain Mapping, 39(10), 3956-3971. https://doi.org/10.1002/hbm.24223 doi: 10.1002/hbm.24223 URL pmid: 30024084
[86]	Szkudlarek, E., & Brannon, E. M.(2018). Approximate arithmetic training improves informal math performance in low achieving preschoolers. Frontiers in Psychology, 9, 606. https://doi.org/10.3389/fpsyg.2018.00606 doi: 10.3389/fpsyg.2018.00606 URL pmid: 29867624
[87]	Vallentin, D., & Nieder, A.(2008). Behavioral and prefrontal representation of spatial proportions in the monkey. Current Biology, 18(18), 1420-1425. https://doi.org/10.1016/j.cub.2008.08.042 doi: 10.1016/j.cub.2008.08.042 URL pmid: 18804374
[88]	vanMarle, K., Chu, F. W., Li, Y. R., & Geary, D. C.(2014). Acuity of the approximate number system and preschoolers' quantitative development. Developmental Science, 17(4), 492-505. https://doi.org/10.1111/desc.12143 doi: 10.1111/desc.2014.17.issue-4 URL
[89]	vanMarle, K., Mou, Y., & Seok, J. H.(2016). Analog magnitudes support large number ordinal judgments in infancy. Perception, 45(1-2), 32-43. https://doi.org/10.1177/0301006615602630 doi: 10.1177/0301006616671273 URL
[90]	Wang, J. J., Halberda, J., & Feigenson, L.(2020). Emergence of the link between the approximate number system and symbolic math ability. Child Development, Advance online publication. https://doi.org/10.1111/cdev.13454
[91]	Xu, F., & Garcia, V.(2008). Intuitive statistics by 8-month- old infants. Proceedings of the National Academy of Sciences of the United States of America, 105(13), 5012-5015. https://doi.org/10.1073/pnas.0704450105
[92]	Zhang, L., Wang, Q., Lin, C., Ding, C., & Zhou, X.(2013). An ERP study of the processing of common and decimal fractions: How different they are. PLOS ONE, 8(7), e69487. https://doi.org/10.1371/journal.pone.0069487 doi: 10.1371/journal.pone.0069487 URL

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