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Which Type of Rational Numbers Should Students Learn First?

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Abstract

Many children and adults have difficulty gaining a comprehensive understanding of rational numbers. Although fractions are taught before decimals and percentages in many countries, including the USA, a number of researchers have argued that decimals are easier to learn than fractions and therefore teaching them first might mitigate children’s difficulty with rational numbers in general. We evaluate this proposal by discussing evidence regarding whether decimals are in fact easier to understand than fractions and whether teaching decimals before fractions leads to superior learning. Our review indicates that decimals are not generally easier to understand than fractions, though they are easier on some tasks. Learners have similar difficulty in understanding fraction and decimal magnitudes, arithmetic, and density, as well as with converting from either notation to the other. There was too little research on knowledge of percentages to include them in the comparisons or to establish the ideal order of instruction of the three types of rational numbers. Although existing research is insufficient to determine the best sequence for teaching the three rational number formats, we recommend several types of research that could help in addressing the issue in the future.

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References

  • Australian Curriculum and Assessment Reporting Authority. (2014). Foundation to year 10 curriculum: mathematics. http://www.australiancurriculum.edu.au/mathematics/curriculum/f-10?layout=1. Accessed 25 February 2017.

  • Behr, M. J., Harel, G., Post, T. R., & Lesh, R. (1992). Rational number, ratio, and proportion. In Handbook of research on mathematics teaching and learning (pp. 296–333). New York: Macmillan.

    Google Scholar 

  • Behr, M. J., Lesh, R., Post, T. R., & Silver, E. A. (1983). Rational number concepts. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 91–125). New York: Academic Press http://www.discoveryeducation.com/teachers/free-lesson-plans/rational-number-concepts.cfm.

    Google Scholar 

  • Behr, M. J., Wachsmuth, I., Post, T. R., & Lesh, R. (1984). Order and equivalence of rational numbers: a clinical teaching experiment. Journal for Research in Mathematics Education, 15(5), 323–341. doi:10.2307/748423.

    Article  Google Scholar 

  • Blair, K., Rosenberg-Lee, M., Tsang, J. M., Schwartz, D. L., & Menon, V. (2012). Beyond natural numbers: negative number representation in parietal cortex. Frontiers in Human Neuroscience, 6, 1–17. doi:10.3389/fnhum.2012.00007.

    Article  Google Scholar 

  • Booth, J. L., & Newton, K. J. (2012). Fractions: could they really be the gatekeeper’s doorman? Contemorary Educational Psychology, 37, 247–253.

    Article  Google Scholar 

  • Booth, J. L., Newton, K. J., & Twiss-Garrity, L. K. (2014). The impact of fraction magnitude knowledge on algebra performance and learning. Journal of Experimental Child Psychology, 118(1), 110–118. doi:10.1016/j.jecp.2013.09.001.

    Article  Google Scholar 

  • Braithwaite, D. W., & Siegler, R. S. (2017). Developmental changes in the whole number bias. Developmental Science. doi:10.1111/desc.12541.

  • Carpenter, T. P., Coburn, T. G., Reys, R. E., & Wilson, J. W. (1975). Results and implications of the NAEP mathematics assessment: secondary school. The Mathematics Teacher, 68(6), 453–470.

    Google Scholar 

  • Carpenter, T. P., Corbitt, M. K., Kepner, H. S., Lindquist, M. M., & Reys, R. E. (1981). Decimals: results and implications from national assessment. The Arithmetic Teacher, 28(8), 34–37.

    Google Scholar 

  • Carpenter, T. P., Kepner, H., Corbitt, M. K., Lindquist, M. M., & Reys, R. E. (1980). Results and implications of the second NAEP mathematics assessment: elementary school. The Arithmetic Teacher, 27(8), 10–47.

    Google Scholar 

  • Charalambous, C. Y., & Pitta-Pantazi, D. (2007). Drawing on a theoretical model to study students’ understandings of fractions. Educational Studies in Mathematics, 64(3), 293–316. doi:10.1007/s10649-006-9036-2.

    Article  Google Scholar 

  • Common Core State Standards Initiative (2010). Common core state standards for mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf. Accessed 25 February 2017.

  • Confrey, J., & Smith, E. (1994). Exponential functions, rates of change, and the multiplicative unit. Educational Studies in Mathematics, 26(2), 135–164.

    Article  Google Scholar 

  • Cramer, K. A., Post, T. R., & DelMas, R. C. (2002). Initial fraction learning by fourth- and fifth-grade students: a comparison of the effects of using commercial curricula with the effects of using the rational number project curriculum. Journal for Research in Mathematics Education, 33(2), 111–144. doi:10.2307/749646.

    Article  Google Scholar 

  • Department for Education (2013). The National Curriculum in England: key stages 1 and 2 framework document. https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/425601/PRIMARY_national_curriculum.pdf. Accessed 25 February 2017.

  • Desmet, L., Grégoire, J., & Mussolin, C. (2010). Developmental changes in the comparison of decimal fractions. Learning and Instruction, 20(6), 521–532. doi:10.1016/j.learninstruc.2009.07.004.

    Article  Google Scholar 

  • DeWolf, M., Bassok, M., & Holyoak, K. J. (2015a). From rational numbers to algebra: separable contributions of decimal magnitude and relational understanding of fractions. Journal of Experimental Child Psychology, 133, 72–84. doi:10.1016/j.jecp.2015.01.013.

    Article  Google Scholar 

  • DeWolf, M., Bassok, M., & Holyoak, K. J. (2015b). Conceptual structure and the procedural affordances of rational numbers: relational reasoning with fractions and decimals. Journal of experimental psychology. General, 144(1), 127–150. doi:10.1037/xge0000034.

    Google Scholar 

  • DeWolf, M., Grounds, M. a., Bassok, M., & Holyoak, K. J. (2014). Magnitude comparison with different types of rational numbers. Journal of Experimental Psychology: Human Perception and Performance, 40(1), 71–82. doi:10.1037/a0032916.

    Google Scholar 

  • Duncan, G. J., Dowsett, C. J., Claessens, A., Magnuson, K., Huston, A. C., Klebanov, P., et al. (2007). School readiness and later achievement. Developmental Psychology, 43(6), 1428–1446. doi:10.1037/0012-1649.43.6.1428.

    Article  Google Scholar 

  • Durkin, K., & Rittle-Johnson, B. (2015). Diagnosing misconceptions: revealing changing decimal fraction knowledge. Learning and Instruction, 37, 21–29. doi:10.1016/j.learninstruc.2014.08.003.

    Article  Google Scholar 

  • Erlwanger, S. (1973). Benny’s conception of rules and answers in IPI mathematics. Journal of Children’s Mathematical Behavior, 1, 7–26.

    Google Scholar 

  • Fischbein, E., Deri, M., Nello, M. S., & Marino, M. S. (1985). The role of implicit models in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education, 16(1), 3–17. doi:10.2307/748969.

    Article  Google Scholar 

  • Fosnot, C. T., & Dolk, M. (2002). Young mathematicians at work, 3: constructing fractions, decimals, and percents. Portsmouth: Heinemann.

    Google Scholar 

  • Ganor-Stern, D. (2013). Are 1/2 and 0.5 represented in the same way? Acta Psychologica, 142(3), 299–307. doi:10.1016/j.actpsy.2013.01.003.

    Article  Google Scholar 

  • Gay, A. S., & Aichele, D. B. (1997). Middle school students’ understanding of number sense related to percent. School Science and Mathematics, 97(1), 27–36. doi:10.1111/j.1949-8594.1997.tb17337.x.

    Article  Google Scholar 

  • Geary, D. C., Hamson, C. O., Chen, G.-P., Liu, F., Hoard, M. K., Salthouse, T. A., et al. (1997). Computational and reasoning abilities in arithmetic: cross-generational change in China and the United States. Psychonomic Bulletin & Review, 4(3), 425–430. doi:10.3758/BF03210805.

    Article  Google Scholar 

  • Geary, D. C., Hoard, M. K., & Bailey, D. H. (2012). Fact retrieval deficits in low achieving children and children with mathematical learning disability. Journal of Learning Disabilities, 45(4), 291–307. doi:10.1177/0022219410392046.

    Article  Google Scholar 

  • Geary, D. C., Salthouse, T. A., Chen, G. P., & Fan, L. (1996). Are East Asian versus American differences in arithmetical ability a recent phenomenon? Developmental Psychology, 32(2), 254–262. doi:10.1037//0012-1649.32.2.254.

    Article  Google Scholar 

  • Gelman, R. (1991). Epigenetic foundations of knowledge structures: initial and transcendent constructions. In S. Carey & R. Gelman (Eds.), The epigenesis of mind: essays on biology and cognition (pp. 293–322). Hillsdale: Erlbaum.

    Google Scholar 

  • Giannakoulias, E., Souyoul, A., & Zachariades, T. (2007). Students’ thinking about fundamental real numbers properties. In D. Pitta-Pantazi, & G. Philippou (Eds.), Proceedings of the Fifth Congress of the European Society for Research in Mathematics Education. Cyrus: ERME, Department of Education, University of Cyprus (pp. 416–425).

  • Graeber, A. O., & Tirosh, D. (1990). Insights fourth and fifth graders bring to multiplication and division with decimals. Educational Studies in Mathematics, 21(6), 565–588. doi:10.1007/BF00315945.

    Article  Google Scholar 

  • Guiler, W. S. (1946a). Difficulties in percentage encountered by ninth-grade pupils. The Elementary School Journal, 46(10), 563–573 http://www.jstor.org/stable/998710.

    Article  Google Scholar 

  • Guiler, W. S. (1946b). Difficulties encountered in percentage by college freshmen. Journal of Educational Research, 40(2), 81–95 http://www.jstor.org/stable/27528757.

    Article  Google Scholar 

  • Handel, M. (2016). What do people do at work? A profile of U.S. jobs from the survey of workplace Skills, Technology, and Management Practices (STAMP). Journal for Labour Market Research, 49(2), 177–197. doi:10.1007/s12651-016-0213-1.

    Article  Google Scholar 

  • Hanson, S. A., & Hogan, T. P. (2000). Computational estimation skill of college students. Journal for Research in Mathematics Education, 31(4), 483–499. doi:10.2307/749654.

    Article  Google Scholar 

  • Hecht, S. A. (1998). Toward an information-processing account of individual differences in fraction skills. Journal of Educational Psychology, 90(3), 545–559. doi:10.1037/0022-0663.90.3.545.

    Article  Google Scholar 

  • Hiebert, J., & Wearne, D. (1983, April). Students’ conceptions of decimal numbers. Paper presented at the annual meeting of the American Educational Research Association, Montreal, Quebec, Canada.

  • Hiebert, J., & Wearne, D. (1985). A model of students’ decimal computation procedures. Cognition and Instruction, 2(3), 175–205. doi:10.1080/07370008.1985.9648916.

    Article  Google Scholar 

  • Hiebert, J., & Wearne, D. (1986). Procedures over concepts: the acquisition of decimal number knowledge. In J. Hiebert (Ed.), Conceptual and procedural knowledge: the case of mathematics (pp. 199–223). Hillsdale: Erlbaum.

    Google Scholar 

  • Hurst, M., & Cordes, S. (2016). Rational-number comparison across notation: fractions, decimals, and whole numbers. Journal of Experimental Psychology: Human Perception and Performance, 42(2), 281–293. doi:10.1037/xhp0000140.

    Google Scholar 

  • Iuculano, T., & Butterworth, B. (2011). Understanding the real value of fractions and decimals. Quarterly Journal of Experimental Psychology, 64(11), 2088–2098. doi:10.1080/17470218.2011.604785.

    Article  Google Scholar 

  • Johnson, J. T. (1956). Decimal versus common fractions. The Arithmetic Teacher, 3(5), 201–206.

    Google Scholar 

  • 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. doi:10.1016/j.jecp.2013.02.001.

    Article  Google Scholar 

  • Kalchman, M., Moss, J., & Case, R. (2001). Psychological models for the development of mathematical understanding: rational numbers and functions. In S. Carver & D. Klahr (Eds.), Cognition and instruction: twenty-five years of progress (pp. 1–38). Mahwah: Erlbaum.

    Google Scholar 

  • Kallai, A. Y., & Tzelgov, J. (2011). When meaningful components interrupt the processing of the whole: the case of fractions. ACTPSY, 139(2), 358–369. doi:10.1016/j.actpsy.2011.11.009.

    Google Scholar 

  • Kellman, P. J., Massey, C., Roth, Z., Burke, T., Zucker, J., Saw, A., et al. (2008). Perceptual learning and the technology of expertise studies in fraction learning and algebra. Pragmatics & Cognition, 16(2), 356–405. doi:10.1075/p&c.16.2.07kel.

    Article  Google Scholar 

  • Kieren, T. E. (1976). On the mathematical, cognitive and instructional foundations of rational numbers. In R. Lesh (Ed.), Number and measurement: papers from a research workshop (pp. 101–144). Columbus, Oh: ERIC/SMEAC.

  • Kieren, T. E. (1980). The rational number construct: its elements and mechanisms. In Recent research on number learning (pp. 125–150).

  • Kloosterman, P. (2010). Mathematics skills of 17-year-olds in the United States: 1978 to 2004. Journal for Research in Mathematics Education, 41(1), 20–51. doi:10.2307/40539363.

    Google Scholar 

  • Kloosterman, P. (2012). Mathematics performance of 13-year olds in the United States: 1978 to 2004., 1, 1–30. doi:10.1017/CBO9781107415324.004.

  • Koedel, C., & Tyhurst, E. (2012). Math skills and labor-market outcomes: evidence from a resume-based field experiment. Economics of Education Review, 31(1), 131–140. doi:10.1016/j.econedurev.2011.09.006.

    Article  Google Scholar 

  • Kouba, V. L., Brown, C. A., Carpenter, T. P., Lindquist, M. M., Silver, E. A., & Swafford, J. O. (1988). Results of the fourth NAEP assessment of mathematics: number, operations, and word problems. The Arithmetic Teacher, 35(8), 14–19.

    Google Scholar 

  • Lachance, A., & Confrey, J. (1995). Introducing fifth graders to decimal notation through ratio and proportion. Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education.

  • Lai, M. Y., & Murray, S. (2014). What do error patterns tell us about Hong Kong Chinese and Australian students’ understanding of decimal numbers? International Journal for Mathematics Teaching & Learning. http://www.cimt.org.uk/journal/lai2.pdf. Accessed 4 Jan 2017.

  • Lamon, S. J. (2012). Teaching fractions and ratios for understanding: essential content knowledge and instructional strategies for teachers. New York: Routledge.

    Google Scholar 

  • Lembke, L. O., & Reys, B. J. (1994). The development of, and interaction between, intuitive and school-taught ideas about percent. Journal for Research in Mathematics Education, 25(3), 237–259. doi:10.2307/749290.

    Article  Google Scholar 

  • Lortie-Forgues, H., & Siegler, R. S. (2017). Conceptual knowledge of decimal arithmetic. Journal of Educational Psychology, 109(3), 374–386. doi:10.1037/edu0000148.

    Article  Google Scholar 

  • Lortie-Forgues, H., Tian, J., & Siegler, R. S. (2015). Why is learning fraction and decimal arithmetic so difficult? Developmental Review, 38, 201–221. doi:10.1016/j.dr.2015.07.008.

    Article  Google Scholar 

  • Ma, L. (1999). Knowing and teaching elementary mathematics: teachers’ understanding of fundamental mathematics in China and the United States. Mahwah: Lawrence Erlbaum Associates. doi:10.5860/CHOICE.37-2294.

    Google Scholar 

  • Mack, N. K. (1995). Confounding whole-number and fraction concepts when building an informal knowledge. Journal for Research in Mathematics Education, 26(5), 422–441. doi:10.2307/749431.

    Article  Google Scholar 

  • Martin, W. G., Strutchens, M. E., & Elliott, P. C. (Eds.) (2007). The learning of mathematics, 69th NCTM yearbook. Reston, VA: National Council of Teachers of Mathematics.

  • McCloskey, M. (2007). Quantitative literacy and developmental dyscalculias. In D. B. Berch & M. M. Mazzocco (Eds.), Why is math so hard for some children? The nature and origins of mathematical learning difficulties and disabilities (pp. 415–429). Baltimore: Paul H. Brookes Publishing.

    Google Scholar 

  • Meert, G., Grégoire, J., & Noël, M. (2009). Rational numbers: componential versus holistic representation of fractions in a magnitude comparison task. The Quarterly Journal of Experimental Psychology, 62(8), 1598–1616. doi:10.1080/17470210802511162.

    Article  Google Scholar 

  • Moss, J. (1997). Developing children’s rational number sense: a new approach and an experimental program. University of Toronto.

  • Moss, J., & Case, R. (1999). Developing children’s understanding of the rational numbers: a new model and an experimental curriculum. Journal for Research in Mathematics Education, 30(2), 122. doi:10.2307/749607.

    Article  Google Scholar 

  • Nesher, P., & Peled, I. (1986). Shifts in reasoning. Educational Studies in Mathematics, 17(1), 67–79. doi:10.1007/BF00302379.

    Article  Google Scholar 

  • Newton, K. J. (2008). An extensive analysis of preservice elementary teachers’ knowledge of fractions. American Educational Research Association, 45(4), 1080–1110. doi:10.3102/0002831208320851.

    Article  Google Scholar 

  • Ni, Y., & 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. doi:10.1207/s15326985ep4001_3.

    Article  Google Scholar 

  • O’Connor, M. C. (2001). “Can any fraction be turned into a decimal?” A case study of a mathematical group discussion. Educational Studies in Mathematics, 46(1), 143–185.

    Article  Google Scholar 

  • Obersteiner, A., Van Dooren, W., Van Hoof, J., & Verschaffel, L. (2013). The natural number bias and magnitude representation in fraction comparison by expert mathematicians. Learning and Instruction, 28, 64–72. doi:10.1016/j.learninstruc.2013.05.003.

    Article  Google Scholar 

  • Pagni, D. (2004). Fractions and decimals. Australian Mathematics Teacher, 60(4), 28–30.

    Google Scholar 

  • Parker, M., & Leinhardt, G. (1995). Percent: a privileged proportion. Review of Educational Research, 65(4), 421–481. doi:10.3102/00346543065004421.

    Article  Google Scholar 

  • Resnick, L. B., Nesher, P., Leonard, F., Magone, M., Omanson, S., & Peret, I. (1989). Conceptual bases of arithmetic errors: the case of decimal fractions. Journal for Research in Mathematics Education, 20(1), 8–27. doi:10.2307/749095.

    Article  Google Scholar 

  • Ritchie, S. J., & Bates, T. C. (2013). Enduring links from childhood mathematics and reading achievement to adult socioeconomic status. Psychological Science, 24(7), 1301–1308. doi:10.1177/0956797612466268.

    Article  Google Scholar 

  • Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: an iterative process. Journal of Educational Psychology, 93(2), 346–362. doi:10.1037/0022-0663.93.2.346.

    Article  Google Scholar 

  • Sackur-Grisvard, C., & Léonard, F. (1985). Intermediate cognitive organizations in the process of learning a mathematical concept: the order of positive decimal numbers. Cognition and Instruction, 2(2), 157–174. doi:10.1207/s1532690xci0202_3.

    Article  Google Scholar 

  • Schaie, K. W. (1993). The Seattle longitudinal studies of adult intelligence. Current Directions in Psychological Science, 2(6), 171–175.

    Article  Google Scholar 

  • Sformo, T. (2008). Practical problems in mathematics: for automotive technicians. Independence: Cengage Learning.

    Google Scholar 

  • Siegler, R. S., Carpenter, T. P., Fennell, F., Geary, D. C., Lewis, J., Okamoto, Y., et al. (2010). Developing effective fractions instruction for kindergarten through 8th grade: a practice guide. Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. http://eric.ed.gov/ERICWebPortal/recordDetail?accno=ED512043

  • Siegler, R. S., Duncan, G. J., Davis-Kean, P. E., Duckworth, K., Claessens, A., Engel, M., et al. (2012). Early predictors of high school mathematics achievement. Psychological Science, 23(7), 691–697. doi:10.1177/0956797612440101.

    Article  Google Scholar 

  • Siegler, R. S., & Lortie-Forgues, H. (2015). Conceptual knowledge of fraction arithmetic. Journal of Educational Psychology, 107(3), 909–918. doi:10.1037/edu0000025.

    Article  Google Scholar 

  • Siegler, R. S., & Pyke, A. A. (2013). Developmental and individual differences in understanding of fractions. Developmental Psychology, 49(10), 1994–2004. doi:10.1037/a0031200.

    Article  Google Scholar 

  • Siegler, R. S., Thompson, C. A., & Schneider, M. (2011). An integrated theory of whole number and fractions development. Cognitive Psychology, 62(4), 273–296. doi:10.1016/j.cogpsych.2011.03.001.

    Article  Google Scholar 

  • Silver, E. A. (1983). Probing young adults’ thinking about rational numbers. Focus on Learning Problems in Mathematics, 5, 105–117.

    Google Scholar 

  • Smith, M. S., Silver, E. A., & Stein, M. K. (2005). Improving instruction in rational numbers and proportionality (Vol. 1). New York: Teachers College Press.

    Google Scholar 

  • Stafylidou, S., & Vosniadou, S. (2004). The development of students’ understanding of the numerical value of fractions. Learning and Instruction, 14(5), 503–518. doi:10.1016/j.learninstruc.2004.06.015.

    Article  Google Scholar 

  • Steffe, L. P. (2001). A new hypothesis concerning children’s fractional knowledge. Journal of Mathematical Behavior, 20(3), 267–307. doi:10.1016/S0732-3123(02)00075-5.

    Article  Google Scholar 

  • Stigler, J. W., Givvin, K. B., Thompson, B. J., Stigler, J. W., & Thompson, B. J. (2010). What community college developmental mathematics students understand about mathematics. Mathematics Teacher, 1(3), 4–16 http://statlit.org/pdf/2009CarnegieFoundation-Developmental-Math-CC-Students-Understand.pdf.

    Google Scholar 

  • Sweeney, E. S., & Quinn, R. J. (2000). Concentration: connecting fractions, decimals & percents. Mathematics Teaching in the Middle Schooliddle School, 5, 324–328 http://search.proquest.com/docview/231158993?accountid=9902.

    Google Scholar 

  • Tian, J., & Siegler, R. S. (2017, April). Influence of number of digits on rational number magnitude understanding. Poster presented at the biennial meeting of the Society for Research in Child Development, Austin, TX.

  • Tirosh, D., Fischbein, E., Graeber, A. O., & Wilson, J. W. (1999). Prospective elementary teachers’ conceptions of rational numbers, 1–17.

  • Torbeyns, J., Schneider, M., Xin, Z., & Siegler, R. S. (2015). Bridging the gap: fraction understanding is central to mathematics achievement in students from three different continents. Learning and Instruction, 37, 5–13. doi:10.1016/j.learninstruc.2014.03.002.

    Article  Google Scholar 

  • Vamvakoussi, X., Christou, K. P., Mertens, L., & Van Dooren, W. (2011). What fills the gap between discrete and dense? Greek and Flemish students’ understanding of density. Learning and Instruction, 21(5), 676–685. doi:10.1016/j.learninstruc.2011.03.005.

    Article  Google Scholar 

  • Vamvakoussi, X., Van Dooren, W., & Verschaffel, L. (2012). Naturally biased? In search for reaction time evidence for a natural number bias in adults. Journal of Mathematical Behavior, 31(3), 344–355. doi:10.1016/j.jmathb.2012.02.001.

    Article  Google Scholar 

  • Vamvakoussi, X., & Vosniadou, S. (2007). How many numbers are there in a rational numbers interval? Constraints, synthetic models and the effect of the number line. In S. Vosniadou, A. Baltas, & X. Vamvakoussi (Eds.), Reframing the conceptual change approach in learning and instruction (pp. 265–282). Oxford: Elsevier.

    Google Scholar 

  • Vamvakoussi, X., & Vosniadou, S. (2010). How many decimals are there between two fractions? Aspects of secondary school students’ understanding of rational numbers and their notation. Cognition and Instruction, 28(2), 181–209. doi:10.1080/07370001003676603.

    Article  Google Scholar 

  • Varma, S., & Karl, S. R. (2013). Understanding decimal proportions: discrete representations, parallel access, and privileged processing of zero. Cognitive Psychology, 66(3), 283–301. doi:10.1016/j.cogpsych.2013.01.002.

    Article  Google Scholar 

  • Wang, Y. Q., & Siegler, R. S. (2013). Representations of and translation between common fractions and decimal fractions. Chinese Science Bulletin, 58(36), 4630–4640. doi:10.1007/s11434-013-6035-4.

    Article  Google Scholar 

  • 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). doi:10.1371/journal.pone.0069487.

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Acknowledgements

The research reported here was supported in part by the Institute of Education Sciences, US Department of Education, through Grants R305A150262 and R324C100004:84.324C, Subaward 23149 to Carnegie Mellon University, in addition to the Teresa Heinz Chair at Carnegie Mellon University and the Siegler Center for Innovative Learning and Advanced Technology Center, Beijing Normal University. The opinions expressed are those of the authors and do not represent views of the Institute or the US Department of Education.

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  1. Department of Psychology, Carnegie Mellon University, Baker Hall, 5000 Forbes Ave, Pittsburgh, PA, 15213, USA

    Jing Tian & Robert S. Siegler

  2. Siegler Center for Innovative Learning and Advanced Technology Center, Beijing Normal University, Beijing, 100875, China

    Jing Tian & Robert S. Siegler

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Tian, J., Siegler, R.S. Which Type of Rational Numbers Should Students Learn First?. Educ Psychol Rev 30, 351–372 (2018). https://doi.org/10.1007/s10648-017-9417-3

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