The development of rational number knowledge: Old topic, new insights

Learning and Instruction

Volumn 37, Issue , 2015, Pages 50-55

  Vamvakoussi, Xenia ^a  

^a UNIVERSITY OF IOANNINA   (Greece)

Author keywords

Rational number knowledge; Whole natural number bias

Indexed keywords

EID: 84940009238     PISSN: 09594752     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.learninstruc.2015.01.002     Document Type: Note

References (40)

1. Behr M., Harel G., Post T., Lesh R. Units of quantity: a conceptual basis common to additive and multiplicative structures. The development of multiplicative reasoning in the learning of mathematics 1994, 123-180. SUNY Press, Albany, NY. G. Harel, J. Confrey (Eds.).
2. 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(6):1410-1419. 10.1037/0096-1523.33.6.1410.
3. Brousseau G. Theory of didactical situations in mathematics 2002, (N. Balacheff, M. Cooper, R. Sutherland, & V. Warfield, Eds. and Trans.), Kluwer Academic Publishers, New York, USA.
4. Christou K.P., Vosniadou S. What kinds of numbers do students assign to literal symbols? Aspects of the transition from arithmetic to algebra. Mathematical Thinking and Learning 2012, 14(1):1-27. 10.1080/10986065.2012.625074.
5. Confrey J., Maloney A.P., Nguyen K.H., Mojica G., Myers M. Equipartitioning/splitting as a foundation of rational number reasoning using learning trajectories. Proceedings of the 33rd conference of the international group for the psychology of mathematics education: In search for theories in mathematics education 2009, Vol. 2:345-352. PME, Thessaloniki, Greece.
6. DeWolf M., Vosniadou S. The representation of fraction magnitudes and the whole number bias reconsidered. Learning and Instruction 2015, 37:39-49. http://dx.doi.org/10.1016/j.learninstruc.2014.07.002.
7. Fischbein E. Intuition in science and mathematics: An educational approach 1987, Reidel, Dordrecht, The Netherlands.
8. Gallistel C.R., Gelman R. Nonverbal numerical cognition: from reals to integers. Trends in Cognitive Science 2000, 4(2):59-65. 10.1016/S1364-6613(99)01424-2.
9. The development of multiplicative reasoning in the learning of mathematics 1994, State University of New York Press, Albany, NY. G. Harel, J. Confrey (Eds.).
10. Jacob N.J., Vallentin D., Nieder A. Relating magnitudes: the brain's code for proportions. Trends in Cognitive Science 2012, 16:157-166. 10.1016/j.tics.2012.02.002.
11. Kallai A.Y., Tzelgov J. Ageneralized fraction: an entity smaller than one on the mental number line. Journal of Experimental Psychology: Human Perception and Performance 2009, 35:1845-1864. 10.1037/a0016892.
12. Kallai A.Y., Tzelgov J. When meaningful components interrupt the processing of the whole: the case of fractions. Acta Psychologica 2011, 139:358-369. 10.1016/j.actpsy.2011.11.009.
13. Kelley D., Rittle-Johnson B. Diagnosing misconceptions: Revealing changing decimal fraction knowledge. Learning and Instruction 2015, 37:21-29. http://dx.doi.org/10.1016/j.learninstruc.2014.08.003.
14. Kieren T.E. Rational and fractional numbers as mathematical and personal knowledge: implications for curriculum and instruction. Analysis of arithmetic for mathematics teaching 1992, 323-371. Lawrence Erlbaum, Hillsdale, NJ. G. Leinhardt, R. Putnam, R.A. Hattrup (Eds.).
15. Kilpatrick J., Swafford J., Findell B. Adding it up. Helping children learn mathematics 2001, National Academy Press, Washington, DC.
16. Lamon S.J. Teaching fractions and ratios for understanding 2006, Lawrence Erlbaum Associates, Mahwah, New Jersey.
17. Le Corre M., Carey S. One, two, three, four, nothing more: an investigation of the conceptual sources of the verbal counting principles. Cognition 2007, 105:395-438. 10.1016/j.cognition.2006.10.005.
18. McMullen J., Laakkonen E., Hannula-Sormunen M., Lehtinen E. Modeling the developmental trajectories of rational number concept(s). Learning and Instruction 2015, 37:14-20. http://dx.doi.org/10.1016/j.learninstruc.2013.12.004.
19. 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(3):284-292. 10.1016/j.actpsy.2010.07.014.
20. 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(3):244-259. 10.1016/j.jecp.2010.04.008.
21. Merenluoto K., Lehtinen E. Number concept and conceptual change: towards a systemic model of the processes of change. Learning and Instruction 2004, 14:519-536. 10.1016/j.learninstruc.2004.06.016.
22. Moss J. Pipes, tubes, and beakers: new approaches to teaching the rational-number system. How students learn: Mathematics in the classroom 2005, 121-162. National Academic Press, Washington, DC. M.S. Donovan, J.D. Bransford (Eds.).
23. Ni Y. How valid is it to use number lines to measure children's conceptual knowledge about rational number?. Educational Psychology 2000, 20(2):139-152. 10.1080/713663716.
24. Ni Y., Zhou Y.-D. Teaching and learning fraction and rational numbers: the origins and implications of whole number bias. Educational Psychologist 2005, 40(1):27-52. 10.1207/s15326985ep4001_3.
25. Nunes T., Bryant P. Children doing mathematics 1996, Blackwell, Oxford, UK.
26. Obersteiner A., Van Dooren W., Van Hoof J., Verschaffel L. The natural number bias and magnitude representation in fraction comparison by expert mathematicians. Learning and Instruction 2013, 28:64-72. 10.1016/j.learninstruc.2013.05.003.
27. 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.
28. Siegler R.S., Thompson C.A., Schneider M. An integrated theory of whole number and fractions development. Cognitive Psychology 2011, 62(4):273-296. 10.1016/j.cogpsych.2011.03.001.
29. Sophian C. Aprospective developmental perspective on early mathematics instruction. Engaging young children in mathematics: Standards for early childhood mathematics 2004, 253-256. Lawrence Erlbaum Associates, Mahwah, NJ. D.H. Clements, J. Sarama, A.-M. DiBiase (Eds.).
30. Sophian C. Precursors to number: equivalence relations, less-than and greater-than relations, and units. Behavioral and Brain Sciences 2008, 31:670-671. 10.1017/S0140525X08005566.
31. Steffe L.P., Olive J. Children's fractional knowledge 2010, Springer, New York, US.
32. Torbeyns J., Schneider M., Xin Z., Siegler R.S. Bridging the gap: Fraction understanding is central to mathematics achievement in students from three different continents. Learning and Instruction 2015, 37:5-13. http://dx.doi.org/10.1016/j.learninstruc.2014.03.002.
33. Tzelgov J., Ganor-Stern D., Kallai A., Pinhas M. Primitives and non-primitives of numerical representations. Oxford handbook of mathematical cognition 2013, 55-66. Oxford University Press, Oxford, England. R. Cohen Kadosh, A. Dowker (Eds.).
34. 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(3):344-355. 10.1016/j.jmathb.2012.02.001.
35. 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(2):181-209. 10.1080/07370001003676603.
36. Vamvakoussi X., Vosniadou S. Bridging the gap between the dense and the discrete: the number line and the "rubber line" bridging analogy. Mathematical Thinking and Learning 2012, 14:265-284.
37. Vamvakoussi X., Vosniadou S., Van Dooren W. The framework theory approach applied to mathematics learning. International handbook of research on conceptual change 2013, 305-321. Routledge, New York, US. 2nd ed. S. Vosniadou (Ed.).
38. 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(2):154-164.
39. Van Hoof J., Vandewalle J., Verschaffel L., Van Dooren W. In search for the natural number bias in secondary school students' interpretation of the effect of arithmetical operations. Learning and Instruction 2015, 37:30-38. http://dx.doi.org/10.1016/j.learninstruc.2014.03.004.
40. Vergnaud G. Multiplicative conceptual field: what and why?. The development of multiplicative reasoning in the learning of mathematics 1994, 41-60. State University of New York Press, Albany, NY. G. Harel, J. Confrey (Eds.).