Abstract
The present study cross-sectionally investigated proportional reasoning abilities in 5- to 9-year-old children (n = 185) before they received instruction in proportional reasoning. This study addressed two important aspects of the development of proportional reasoning that remain unclear in the current literature: (1) the age range in which it develops and (2) the influence of the nature of the quantities (discrete or continuous) on children’s performance. Three proportional reasoning tasks (i.e., one with two discrete quantities, one with a discrete and a continuous quantity, and one with two continuous quantities) were used. A two-step cluster analysis was conducted on the groups of children based on qualitative differences in understanding. Six different early stages of proportional reasoning were revealed, showing differences in understanding depending on the nature of the quantities involved and which quantity was unknown. The development of proportional reasoning starts at a very early age but it is not yet fully mastered at the age of 9.
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Acknowledgments
The authors would like to thank all participating schools, children, and families. We would also like to show our gratitude to Natasha Gotink and Elien Graindor for assistance with data collection.
Funding statement
This work was supported by the Research Fund KU Leuven under Grant C1/16/001.
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The subjects’ parents have given their written informed consent. The study protocol has been approved by the research institute’s committee on human research.
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Elien Vanluydt. Centre for Instructional Psychology and Technology, KU Leuven, Dekenstraat 2 box 3773, 3000 Leuven, Belgium. E-mail: elien.vanluydt@kuleuven.be
Current themes of research:
Psychology of mathematics education. Early development of core mathematical competencies. Proportional reasoning. Additive reasoning.
Most relevant publications in the field of Psychology of Education:
Vanluydt, E., Verschaffel, L., & Van Dooren, W. (2018). Emergent proportional reasoning: searching for early traces in four-to five-year olds. In E. Bergqvist, M. Österholm, C. Granberg, & L. Sumpter (Eds.). Proceedings of the 42nd conference of the International Group for the Psychology of mathematics education (Vol. 4, pp. 247–254). Umeå, Sweden: PME. Retrieved from https://www.igpme.org/
Tine Degrande. Centre for Instructional Psychology and Technology, KU Leuven, Dekenstraat 2 box 3773, 3000 Leuven, Belgium.
Current themes of research:
Psychology of mathematics education. Proportional reasoning. Additive reasoning. Word problem solving.
Most relevant publications in the field of Psychology of Education:
Degrande, T., Verschaffel, L., & Van Dooren, W. (2018). Beyond additive and multiplicative reasoning abilities: how preference enters the picture. European Journal of Psychology of Education, 33, 559–576. doi:10.1007/s10212-017-0352-y, 4.
Degrande, T., Van Hoof, J., Verschaffel, L., & Van Dooren, W. (2017). Open word problems: taking the additive or the multiplicative road? ZDM Mathematics Education, 50, 91–102, doi:10.1007/s11858-017-0900-6, 1–2.
Degrande, T., Verschaffel, L., & Van Dooren, W. (2017). Spontaneous focusing on quantitative relations: towards a characterisation. Mathematical Thinking and Learning, 19, 260–275. doi: 10.1080/10986065.2017.1365223, 4.
Degrande T., Verschaffel L., Van Dooren W. (2019). To add or to multiply? An investigation of the role ofpreference in children's solutions of word problems. Learning and Instruction, 61, 60–71. doi:10.1016/j.learninstruc.2019.01.002.
Lieven Verschaffel. Centre for Instructional Psychology and Technology, KU Leuven, Dekenstraat 2 box 3773, 3000 Leuven, Belgium.
Current themes of research:
Mathematical word problem solving. Pre-school mathematics education. Elementary arithmetic. Computational estimation. The role of metacognition and affects. Mathematics education.
Most relevant publications in the field of Psychology of Education:
Bojorque, G., Torbeyns, J., Hannula-Sormunen, M., Van Nijlen, D., & Verschaffel, L. (2017). Development of SFON in Ecuadorian kindergartners: contribution of early numerical abilities and quality of mathematics education. European Journal for Psychology of Education, 32, 449–462, 3.
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Wim Van Dooren. Centre for Instructional Psychology and Technology, KU Leuven, Dekenstraat 2 box 3773, 3000 Leuven, Belgium.
Current themes of research:
Mathematical word problem solving. Mathematics education. Statistical reasoning. Conceptual change. Intuitions and biases in reasoning.
Most relevant publications in the field of Psychology of Education:
Van Dooren, W., De Bock, D., Verschaffel, L. (2010). From addition to multiplication … and back. The development of students’ additive and multiplicative reasoning skills. Cognition and Instruction, 28 (3), 360–381
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Van Hoof, J., Verschaffel, L., Van Dooren, W. (2015). Inappropriately applying natural number properties in rational number tasks: characterizing the development of the natural number bias through primary and secondary education. Educational Studies in Mathematics, 90 (1), 39–56.
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.
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Vanluydt, E., Degrande, T., Verschaffel, L. et al. Early stages of proportional reasoning: a cross-sectional study with 5- to 9-year-olds. Eur J Psychol Educ 35, 529–547 (2020). https://doi.org/10.1007/s10212-019-00434-8
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DOI: https://doi.org/10.1007/s10212-019-00434-8