Abstract
This study tested whether second graders use benchmark-based strategies when solving a number line estimation (NLE) task. Participants were assigned to one of three conditions based on the availability of benchmarks provided on the number line. In the bounded condition, number lines were only bounded at both sides by 0 and 200, while the midpoint condition included an additional benchmark at the midpoint and children in the quartile condition were provided with a benchmark at every quartile. First, the inclusion of a midpoint resulted in more accurate estimates around the middle of the number line in the midpoint condition compared to the bounded and, surprisingly, also the quartile condition. Furthermore, the two additional benchmarks in the quartile condition did not yield better estimations around the first and third quartile, because children frequently relied on an erroneous representation of these benchmarks, leading to systematic estimation errors. Second, verbal strategy reports revealed that children in the midpoint condition relied more frequently on the benchmark at the midpoint of the number line compared to the bounded condition, confirming the accuracy data. Finally, the frequency of use of benchmark-based strategies correlated positively with mathematics achievement and tended to correlate positively also with estimation accuracy. In sum, this study is one of the first to provide systematic evidence for children’s use of benchmark-based estimation strategies in NLE with natural numbers and its relationship with children’s NLE performance.
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Notes
We coded the benchmarks as represented by the child even when misrepresented (e.g., when they thought the 25 % benchmark represented 100 instead of 50).
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Acknowledgments
The conduct of this study was supported by grant GOA 2012/010 of the Research Fund KU Leuven, Belgium, and by grant DFG: EB462/1-1 of the German Research Foundation to the fourth author.
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Dominique Peeters. Centre for Instructional Psychology and Technology, KU Leuven, Dekenstraat 2, box 3773, 3000 Leuven, Belgium. E-mail: dominique.peeters@ppw.kuleuven.be
Current themes of research:
Number line estimation. Numerical cognition. Strategy development.
Tine Degrande. Tine Degrande. Centre for Instructional Psychology and Technology, KU Leuven, Dekenstraat 2, box 3773, 3000 Leuven, Belgium.
Current themes of research:
Proportional reasoning. Additive reasoning. Cognitive development. Mathematics. Primary education.
Most relevant publications in the field of Psychology of Education:
Degrande, T., Verschaffel, L., Van Dooren, W. (2014). How do Flemish children solve ‘Greek’ wordproblems? On children’s quantitative analogical reasoning in mathematically neutral word problems. Mediterranean Journal for Research in Mathematics Education, 13(1--2): 57--74.
Mirjam Ebersbach. Centre for Instructional Psychology and Technology, KU Leuven, Dekenstraat 2, box 3773, 3000 Leuven, Belgium; Institut für Psychologie, Universität Kassel, Holländische Str. 36-38, 34127 Kassel, Germany.
Current themes of research:
Cognitive development. Development of information processing. Mathematical knowledge. Implicit and explicit knowledge. Embodiment.
Most relevant publications in the field of Psychology of Education:
Ebersbach, M., & Erz, P. (2014). Symbolic versus non-symbolic magnitude estimations among children and adults. Journal of Experimental Child Development, 128, 52--68. doi:10.1016/j.jecp.2014.06.005.
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Ebersbach, M., Luwel, K., Frick, A., Onghena, P., & Verschaffel, L. (2008). The relationship between the shape of the mental number line and familiarity with numbers in 5- to 9-year old children: Evidence for a segmented linear model. Journal of Experimental Child Psychology, 99, 1--17. doi:10.1016/j.jecp.2007.08.006.
Ebersbach, M., & Resing, W. C. M. (2008). Implicit and explicit knowledge of linear and exponential growth in 5- and 9-year-olds. Journal of Cognition and Development, 9, 286--309. doi:10.1080/15248370802247962.
Lieven Verschaffel. Centre for Instructional Psychology and Technology, KU Leuven, Dekenstraat 2, box 3773, 3000 Leuven, Belgium.
Current themes of research:
Psychology of mathematics education. Number sense. Estimation. Mental and written arithmetic. Arithmetic word problem solving. Rational number knowledge.
Most relevant publications in the field of Psychology of Education:
Verschaffel, L., Luwel, K., Torbeyns, J., & Van Dooren, W. (2009). Conceptualizing, investigating, and enhancing adaptive expertise in elementary mathematics education. European Journal of Psychology of Education, 24, 335--359.
Dewolf, T., Van Dooren, W., & Verschaffel, L. (2011). Upper elementary school children’s understanding and solution of a quantitative word problem inside and outside the mathematics class. Learning and Instruction, 21, 770--780.
Fernández, C., Llinares, S., Van Dooren, W., De Bock, D., Verschaffel, L. (2012). The development of students’ use of additive and proportional methods along primary and secondary school. European Journal of Psychology of Education, 27, 421--438.
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.
Linsen, S., Verschaffel, L., Reynvoet, B., & De Smedt, B. (2015). The association between numerical magnitude processing and mental versus algorithmic multi-digit subtraction in children. Learning and Instruction, 35, 42--50.
Koen Luwel. Centre for Instructional Psychology and Technology, KU Leuven, Dekenstraat 2, box 3773, 3000 Leuven, Belgium; Centre for Educational Research and Development, KU Leuven—Campus Brussels, Warmoesberg 26, 1000 Brussels, Belgium.
Current themes of research:
Strategy choice and strategy development. Estimation skills. Numerical and mathematical cognition. Number sense.
Most relevant publications in the field of Psychology of Education:
Luwel, K., Foustana, A., Papadatos, Y., & Verschaffel, L. (2011). The role of intelligence and feedback in children’s strategy competence. Journal of Experimental Child Psychology, 108, 61--76.
Verschaffel, L., Luwel, K., Torbeyns, J., & Van Dooren, W. (2009). Conceptualizing, investigating, and enhancing adaptive expertise in elementary mathematics education. European Journal of Psychology of Education, 24, 335--359.
Luwel, K., & Verschaffel, L. (2008). Estimation of ‘real’ numerosities in elementary school children. European Journal of Psychology of Education, 23, 319--338.
Ebersbach, M., Luwel, K., Frick, A., Onghena, P., & Verschaffel, L. (2008). The relationship between the shape of the mental number line and familiarity with numbers in 5- to 9-year old children: Evidence for a segmented linear model. Journal of Experimental Child Psychology, 99, 1--17.
Luwel, K., Siegler, R. S., & Verschaffel, L. (2008). A microgenetic study of insightful problem solving. Journal of Experimental Child Psychology, 99, 210--232.
Dominique Peeters and Tine Degrande contributed equally to this work.
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Peeters, D., Degrande, T., Ebersbach, M. et al. Children’s use of number line estimation strategies. Eur J Psychol Educ 31, 117–134 (2016). https://doi.org/10.1007/s10212-015-0251-z
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DOI: https://doi.org/10.1007/s10212-015-0251-z