Bridging the gap: Fraction understanding is central to mathematics achievement in students from three different continents

Learning and Instruction

Volumn 37, Issue , 2015, Pages 5-13

  Torbeyns, Joke ^a   Schneider, Michael ^b   Xin, Ziqiang ^c   Siegler, Robert S ^d,e  

^a UNIVERSITY OF LEUVEN   (Belgium)

^b UNIVERSITY OF TRIER   (Germany)

^c CENTRAL UNIVERSITY OF FINANCE AND ECONOMICS   (China)

^d CARNEGIE MELLON UNIVERSITY   (United States)

^e BEIJING NORMAL UNIVERSITY   (China)

Author keywords

Arithmetic; Cultural differences; Fractions; Integrated theory of numerical development; Magnitude representations

Indexed keywords

EID: 84897158533     PISSN: 09594752     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.learninstruc.2014.03.002     Document Type: Article

References (55)

1. Bailey D.H., Hoard M.K., Nugent L., Geary D.C. Competence with fractions predicts gains in mathematics achievement. Journal of Experimental Child Psychology 2012, 113:447-455. 10.1016/j.jecp.2012.06.004.
2. Bailey D.H., Siegler R.S., Geary D.C. Early predictors of middle school fraction knowledge. Developmental Science 2014, 10.1111/desc.12155.
3. Ball D.L. The mathematical understandings that prospective teachers bring to teacher education. The Elementary School Journal 1990, 90:449-466. 10.1086/461626.
4. Bloch E.D. The real numbers and real analysis 2011, Springer, New York.
5. Booth J.L., Newton K.J. Fractions: could they really be the gatekeeper's doorman?. Contemporary Educational Psychology 2012, 37:247-253. 10.1016/j.cedpsych.2012.07.001.
6. Booth J.L., Siegler R.S. Developmental and individual differences in pure numerical estimation. Developmental Psychology 2006, 42:189-201. 10.1037/0012-1649.41.6.189.
7. Booth J.L., Siegler R.S. Numerical magnitude representations influence arithmetic learning. Child Development 2008, 79:1016-1031. 10.1111/j.1467-8624.2008.01173.x.
8. De Smedt B., Noël M.-P., Gilmore C., Ansari D. The relationship between symbolic and non-symbolic numerical magnitude processing and the typical and atypical development of mathematics: a review of evidence from brain and behavior. Trends in Neuroscience and Education 2013, 2:48-55.
9. De Smedt B., Verschaffel L., Ghesquière P. The predictive value of numerical magnitude comparison for individual differences in mathematics achievement. Journal of Experimental Child Psychology 2009, 103:469-479. 10.1016/j.jecp.2009.01.010.
10. Deloof G. Leerlingvolgsysteem Wiskunde Toetsen 6: Basisboek 2005, Garant, Antwerpen, Belgium.
11. Depaepe F., Torbeyns J., Vermeersch N., Janssens D., Janssen R., Kelchtermans G., et al. Prospective elementary and lower secondary teachers' content and pedagogical content knowledge in the rational number domain 2013, (submitted for publication).
12. DeWolf M., Vosniadou S. The representation of fraction magnitudes and the whole number bias reconsidered 2014, (in this issue).
13. Ebersbach M., Luwel K., Verschaffel L. Comparing apples and pears in studies on magnitude estimations. Frontiers in Psychology 2013, 4:332. 10.3389/fpsyg.2013.00332.
14. Fuchs L.S., Schumacher R.F., Long J., Namkung J., Hamlett C.L., Cirino P.T., et al. Improving at-risk learners' understanding of fractions. Journal of Educational Psychology 2013, 105:683-700. 10.1037/a0032446.
15. Gabriel F., Coché F., Szucs D., Carette V., Rey B., Content A. Developing children's understanding of fractions: an intervention study. Mind, Brain, and Education 2012, 6:137-146. 10.1111/j.1751-228X.2012.01149.x.
16. Geary D.C. Development of mathematical understanding. Handbook of child psychology: Cognition, perception, and language 2006, Vol. 2:777-810. Wiley, Hoboken, NH. 10.1002/9780470147658.chpsy0218.
17. Gielen S., Van Dessel K., De Meyst M., Beringhs S., Crynen M., Luyten B., et al. Peiling wiskunde in de eerste graad van het secundair onderwijs A-stroom - Eindrapport 2010, K.U. Leuven, Centrum voor Onderwijseffectiviteit en -evaluatie, Leuven.
18. Gilmore C.K., Attridge N., Clayton S., Cragg L., Johnson S., Marlow N., et al. Individual differences in inhibitory control, not non-verbal number acuity, correlate with mathematics achievement. PLOS One 2013, 8(6). 10.1371/journal.pone.0067374.
19. Hattie J. Visible learning. A synthesis of over 800 meta-analyses relating to achievement 2009, Routledge, London.
20. Hill H.C., Rowan B., Ball D. Effects of teachers' mathematical knowledge for teaching on student achievement. American Educational Research Journal 2005, 42:371-406. 10.3102/00028312042002371.
21. Holloway I., Ansari D. Domain-specific and domain-general changes in children's development of number comparison. Developmental Science 2008, 11:644-649. 10.1111/j.1467-7687.2008.00712.x.
22. Huber S., Moeller K., Nuerk H.-C. Dissociating number line estimations from underlying numerical representations. The Quarterly Journal of Experimental Psychology 2013, 10.1080/17470218.2013.838974.
23. Krauss S., Brunner M., Kunter M., Baumert J., Blum W., Neubrand M., et al. Pedagogical content knowledge and content knowledge of secondary mathematics teachers. Journal of Educational Psychology 2008, 100:716-725. 10.1037/0022-0663.100.3.716.
24. Leslie A.M., Gelman R., Gallistel C.R. The generative basis of natural number concepts. Trends in Cognitive Sciences 2008, 12:213-218. 10.1016/j.tics.2008.03.004.
25. Lyons I.M., Ansari D., Beilock S.L. Symbolic estrangement: evidence against a strong association between numerical symbols and the quantities they represent. Journal of Experimental Psychology: General 2012, 141:635-641. 10.1037/a0027248.
26. Ma L. Knowing and teaching elementary mathematics: Teachers understanding of fundamental mathematics in China and the United States 1999, Erlbaum, Mahwah, NJ.
27. Merenluoto K., Lehtinen E. Conceptual change in mathematics: understanding the real numbers. Reconsidering conceptual change. Issues in theory and practice 2002, 233-258. Kluwer Academic Publishers, Dordrecht. 10.1007/0-306-47637-1_13. M. Limón, L. Mason (Eds.).
28. Mullis I.V.S., Martin M.O., Foy P. TIMSS 2007 international mathematics report: Findings from IEA's trends in international mathematics and science study at the fourth and eighth grades 2008, with Olson, J. F., Preuschoff, C., Erberber, E., Arora, A., & Galia, J., TIMSS & PIRLS International Study Center, Boston College, Chestnut Hill, MA.
29. Mullis I.V.S., Martin M.O., Foy P., Arora A. TIMSS 2011 international results in mathematics 2012, TIMSS & PIRLS International Study Center, Boston College, Chestnut Hill, MA.
30. Ni Y., Zhou Y.-D. Teaching and learning fraction and rational numbers: the origins and implications of whole number bias. Educational Psychologist 2005, 40:27-52. 10.1207/s15326985ep4001_3.
31. Opfer J.E., Siegler R.S. Representational change and children's numerical estimation. Cognitive Psychology 2007, 55:169-195. 10.1016/j.cogpsych.2006.09.002.
32. Ramani G.B., Siegler R.S. Promoting broad and stable improvements in low-income children's numerical knowledge through playing number board games. Child Development 2008, 79:375-394. 10.1111/j.1467-8624.2007.01131.x.
33. Ramani G.B., Siegler R.S., Hitti A. Taking it to the classroom: number board games as a small group learning activity. Journal of Educational Psychology 2012, 104:661-672. 10.1037/a0028995.
34. Sasanguie D., Defever E., Van den Bussche E., Reynvoet B. The reliability of and the relation between non-symbolic numerical distance effects in comparison, same-different judgments and priming. Acta Psychologica 2011, 136:73-80. 10.1016/j.actpsy.2010.10.004.
35. Sasanguie D., Göbel S., Moll K., Smets K., Reynvoet B. Acuity of the approximate number sense, symbolic number comparison or mapping numbers onto space: what underlies mathematics achievement?. Journal of Experimental Child Psychology 2013, 114:418-431. 10.1016/j.jecp.2012.10.012.
36. Sasanguie D., Reynvoet B. Number comparison and number line estimation rely on different mechanisms. Psychologica Belgica 2013, 53:17-35.
37. Schneider M., Grabner R.H., Paetsch J. Mental number line, number line estimation, and mathematical achievement: their interrelations in grades 5 and 6. Journal of Educational Psychology 2009, 101:359-372. 10.1037/a0013840.
38. Schneider M., Heine A., Thaler V., Torbeyns J., De Smedt B., Verschaffel L., et al. Avalidation of eye movements as a measure of elementary school children's developing number sense. Cognitive Development 2008, 23:424-437. 10.1016/j.cogdev.2008.07.002.
39. Senk S.L., Tatto M.T., Reckase M., Rowley G., Peck R., Bankov K. Knowledge of future primary teachers for teaching mathematics: an international comparative study. ZDM Mathematics Education 2012, 44:307-324. 10.1007/s11858-012-0400-7.
40. Siegler R.S. Improving the numerical understanding of children from low-income families. Child Development Perspectives 2009, 3:118-124. 10.1111/j.1750-8606.2009.00090.x.
41. Siegler R.S., Duncan G.J., Davis-Kean P.E., Duckworth K., Claessens A., Engel M., et al. Early predictors of high school mathematics achievement. Psychological Science 2012, 23:691-697. 10.1177/0956797612440101.
42. Siegler R.S., Fazio L.K., Bailey D.H., Zhou X. Fractions: the new frontier for theories of numerical development. Trends in Cognitive Science 2013, 17:13-19. 10.1016/j.tics.2012.11.004.
43. Siegler R.S., Pyke A.A. Developmental and individual differences in understanding of fractions. Developmental Psychology 2013, 49:1994-2004. 10.1037/a0031200.
44. Siegler R.S., Ramani G.B. Playing linear number board games - but not circular ones - improves low-income preschoolers' numerical understanding. Journal of Educational Psychology 2009, 101:545-560. 10.1037/a0014239.
45. Siegler R.S., Thompson C.A., Schneider M. An integrated theory of whole number and fractions development. Cognitive Psychology 2011, 62:273-296. 10.1016/j.cogpsych.2011.03.001.
46. Stafylidou S., Vosniadou S. The development of students' understanding of the numerical value of fractions. Learning and Instruction 2004, 14:503-518. 10.1016/j.learninstruc.2004.06.015.
47. Stevenson H.W., Stigler J.W. The learning gap. Why our schools are failing and what we can learn from Japanese and Chinese education 1992, Free Press, New York.
48. Thompson C.A., Siegler R.S. Linear numerical magnitude representations aid children's memory for numbers. Psychological Science 2010, 21:1274-1281. 10.1177/0956797610378309.
49. Vamvakoussi X., Vosniadou S. How many decimals are there between two fractions? Aspects of secondary school students' understanding of rational numbers and their notation. Cognition and Instruction 2010, 28:181-209. 10.1080/07370001003676603.
50. Verschaffel L., Janssens S., Janssen R. The development of mathematical competence in Flemish preservice elementary school teachers. Teaching and Teacher Education 2005, 21:49-63. 10.1016/j.tate.2004.11.005.
51. Vogel S.E., Grabner R.H., Schneider M., Siegler R.S., Ansari D. Overlapping and distinct brain regions involved in estimating the spatial position of numerical and non-numerical magnitudes: an fMRI study. Neuropsychologia 2013, 51:979-989. 10.1016/j.neuropsychologia.2013.02.001.
52. Wang J., Lin E. Ameta-analysis of comparative studies on Chinese and US students' mathematics performance: implications for mathematics education reform and research. Educational Research Review 2009, 4:177-195. 10.1016/j.edurev.2009.06.003.
53. Wu H. How to prepare students for algebra. American Educator 2001, 25(2):10-17.
54. Wu H. What's sophisticated about elementary mathematics?. American Educator 2009, 33(3):4-14.
55. Wynn K. Do infants have numerical expectations or just perceptual preferences?. Developmental Science 2002, 2:207-209. 10.1111/1467-7687.00221_3.