The natural number bias and magnitude representation in fraction comparison by expert mathematicians

Learning and Instruction

Volumn 28, Issue , 2013, Pages 64-72

  Obersteiner, Andreas ^a,b   Van Dooren, Wim ^a   Van Hoof, Jo ^a   Verschaffel, Lieven ^a  

^a UNIVERSITY OF LEUVEN   (Belgium)

^b TECHNICAL UNIVERSITY OF MUNICH   (Germany)

Author keywords

Componential representation; Holistic representation; Intuitive reasoning; Rational numbers; Whole number bias

Indexed keywords

EID: 84879747887     PISSN: 09594752     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.learninstruc.2013.05.003     Document Type: Article

References (42)

1. Babai R., Brecher T., Stavy R., Tirosh D. Intuitive interference in probabilistic reasoning. International Journal of Science and Mathematics Education 2006, 4:627-639. 10.1007/s10763-006-9031-1.
2. Behr M.J., Wachsmuth I., Post T.R., Lesh R. Order and equivalence of rational numbers: a clinical teaching experiment. Journal of Research in Mathematics Education 1984, 15:323-341.
3. Berch D.B. Making sense of number sense: Implications for children with mathematical disabilities. Journal of Learning Disabilities 2005, 38:333-339. 10.1177/00222194050380040901.
4. Bonato M., Fabbri S., Umiltà C., Zorzi M. The mental representation of numerical fractions: real or integer?. Journal of Experimental Psychology: Human Perception and Performance 2007, 33:1410-1419. 10.1037/0096-1523.33.6.1410.
5. Carpenter T.P., Kepner H., Corbitt M.K., Lindquist M.M., Reys R.E. Results and implications of the second NAEP mathematics assessments: elementary school. Arithmetic Teacher 1980, 2(8):10-13.
6. Clarke D.M., Roche A. Students' fraction comparison strategies as a window into robust understanding and possible pointers for instruction. Educational Studies in Mathematics 2009, 72:127-138. 10.1007/s10649-009-9198-9.
7. Dehaene S. The number sense. How the mind creates mathematics 1997, Oxford University Press, New York.
8. Dehaene S., Akhavein R. Attention, automaticity, and levels of representation in number processing. Journal of Experimental Psychology: Learning, Memory, and Cognition 1995, 21:314-326. 10.1037/0278-7393.21.2.314.
9. Dehaene S., Dupoux E., Mehler J. Is numerical comparison digital? Analogical and symbolic effects in two-digit number comparison. Journal of Experimental Psychology: Human Perception and Performance 1990, 16:626-641. 10.1037/0096-1523.16.3.626.
10. DeWolf M., Vosniadou S. The whole number bias in fraction magnitude comparisons with adults. Proceedings of the 33rd annual conference of the cognitive science society 2011, 1751-1756. Cognitive Science Society, Austin, TX. L. Carlson, C. Hoelscher, T.F. Shipley (Eds.).
11. Faulkenberry T.J., Pierce B.H. Mental representations in fraction comparison. Holistic versus component-based strategies. Experimental Psychology 2011, 58:480-489. 10.1027/1618-3169/a000116.
12. 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.
13. Ganor-Stern D. Fractions but not negative numbers are represented on the mental number line. Acta Psychologica 2012, 139:350-357. 10.1016/j.actpsy.2011.11.008.
14. Ganor-Stern D., Karasik-Rivkin I., Tzelgov J. Holistic representation of unit fractions. Experimental Psychology 2011, 58:201-206. 10.1027/1618-3169/a000086.
15. Gillard E., Van Dooren W., Schaeken W., Verschaffel L. Dual processes in the psychology of mathematics education and cognitive psychology. Human Development 2009, 52:95-108. 10.1159/000202728.
16. Greer B. Helping children develop mathematically. Human Development 2009, 52:148-161. 10.1159/000202731.
17. Hubbard E.M., Piazza M., Pinel P., Dehaene S. Interactions between number and space in parietal cortex. Nature Reviews Neuroscience 2005, 6:435-448. 10.1038/nrn1684.
18. Ischebeck A., Schocke M., Delazer M. The processing and representation of fractions within the brain. An fMRI investigation. NeuroImage 2009, 47:403-413. 10.1016/j.neuroimage.2009.03.041.
19. Iuculano T., Butterworth B. Understanding the real value of fractions and decimals. The Quarterly Journal of Experimental Psychology 2011, 64:2088-2098. 10.1080/17470218.2011.604785.
20. Jordan N.C., Glutting J., Ramineni C. The importance of number sense to mathematics achievement in first and third grades. Learning and Individual Differences 2010, 20:82-88. 10.1016/j.lindif.2009.07.004.
21. Kallai A.Y., Tzelgov J. When meaningful components interrupt the processing of the whole: the case of fractions. Acta Psychologica 2012, 139:358-369. 10.1037/a0016892.
22. Mazzocco M.M.M., Devlin K.T. Parts and 'holes': gaps in rational number sense among children with vs. without mathematical learning disabilities. Developmental Science 2008, 11:681-691. 10.1111/j.1467-7687.2008.00717.x.
23. Meert G., Grégoire J., Noël M.-P. Rational numbers: componential versus holistic representation of fractions in a magnitude comparison task. The Quarterly Journal of Experimental Psychology 2009, 62:1598-1616. 10.1080/17470210802511162.
24. Meert G., Grégoire J., Noël M.-P. Comparing 5/7 and 2/9: adults can do it by accessing the magnitude of the whole fractions. Acta Psychologica 2010, 135:284-292. 10.1016/j.actpsy.2010.07.014.
25. Meert G., Grégoire J., Noël M.-P. Comparing the magnitude of two fractions with common components: which representations are used by 10- and 12-year-olds?. Journal of Experimental Child Psychology 2010, 107:244-259. 10.1016/j.jecp.2010.04.008.
26. Moyer R.S., Landauer T.K. Time required for judgements of numerical inequality. Nature 1967, 215:1519-1520. 10.1038/2151519a0.
27. National Council of Teachers of Mathematics (Ed.) Principles and standards for school mathematics 2000, NCTM, VA.
28. 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.
29. Putt I.J. Preservice teachers ordering of decimal numbers: when more issmaller and less is larger!. Focus on Learning Problems in Mathematics 1995, 17(3):1-15.
30. Rau M.A., Aleven V., Rummel N. Interleaved practice in multi-dimensional learning tasks: which dimension should we interleave?. Learning and Instruction 2013, 23:98-114. 10.1016/j.learninstruc.2012.07.003.
31. Schneider M., Siegler R.S. Representations of the magnitudes of fractions. Journal of Experimental Psychology: Human Perception and Performance 2010, 36:1227-1238. 10.1037/a0018170.
32. Schneider W., Eschman A., Zuccolotto A. E-Prime user's guide 2002, Psychology software Tools Inc, Pittsburgh.
33. Sekuler R., Mierkiewicz D. Children's judgments of numerical inequality. Child Development 1977, 48:630-633.
34. Siegler R.S., Fazio L.K., Bailey D.H., Zhou X. Fractions: the new frontier for theories of numerical development. Trends in Cognitive Sciences 2013, 17:13-19. 10.1016/j.tics.2012.11.004.
35. Sowder J.T. Mental computation and number comparisons: the role in development of number sense and computational estimation. Number concepts and operations in the middle grades 1988, 182-197. Lawrence Erlbaum and National Council of Teachers of Mathematics, Reston, VA. J. Hiebert, M. Behr (Eds.).
36. Stacey K., Helme S., Steinle V., Baturo A., Irwin K., Bana J. Preservice teachers' knowledge of difficulties in decimal numeration. Journal of Mathematics Teacher Education 2001, 4:205-225. 10.1023/A:1011463205491.
37. 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.
38. Szücs D., Goswami U. Educational neuroscience: defining a discipline for the study of mental representations. Mind, Brain, and Education 2007, 1(3):114-127. 10.1111/j.1751-228X.2007.00012.x.
39. Vamvakoussi X., Christou K.P., Mertens L., Van Dooren W. What fills the gap between discrete and dense? Greek and Flemish students' understanding of density. Learning and Instruction 2011, 21:676-685. 10.1016/j.learninstruc.2011.03.005.
40. Vamvakoussi X., Van Dooren W., Verschaffel L. Naturally biased? In search for reaction time evidence for a natural number bias in adults. The Journal of Mathematical Behavior 2012, 31:344-355. 10.1016/j.jmathb.2012.02.001.
41. Vamvakoussi X., Vosniadou S. Understanding the structure of the set of rational numbers: a conceptual change approach. Learning and Instruction 2004, 14:453-467. 10.1016/j.learninstruc.2004.06.013.
42. Van Hoof J., Lijnen T., Verschaffel L., Van Dooren W. Are secondary school students still hampered by the natural number bias? A reaction time study on fraction comparison tasks. Research in Mathematics Education 2013, 15:154-164. 10.1080/14794802.2013.797747.