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Various ways to determine rational number size: an exploration across primary and secondary education

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Abstract

Understanding rational numbers is a complex task for primary and secondary school students. Previous research has shown that a possible reason is students’ tendency to apply the properties of natural numbers (inappropriately) when they are working with rational numbers (a phenomenon called natural number bias). Focusing on rational number comparison tasks, recent research has shown that other incorrect strategies such as gap thinking or reverse bias can also explain these difficulties. The present study aims to investigate students’ different ways of thinking when working on fraction and decimal comparison tasks. The participants were 1,262 primary and secondary school students. A TwoStep Cluster Analysis revealed six different student profiles according to their way of thinking. Results showed that while students’ reasoning based on the properties of natural numbers decreased along primary and secondary school, almost disappearing at the end of secondary school, students’ reasoning based on gap thinking increased along these grades. This result seems to indicate that when students overcome their reliance on natural numbers, they enter a stage of qualitatively different errors before finally reaching the stage of correct understanding.

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Acknowledgments

This research was carried out with the support of Conselleria d’Educació, Investigació, Cultura i Esport (Generalitat Valenciana, Spain) (PROMETEO/2017/135) and with the support of the University of Alicante (UAFPU2018-035).

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Authors and Affiliations

  1. Departamento de Innovación y Formación Didáctica, Facultad de Educación, Universidad de Alicante, Calle Aeroplano, s/n, 03690, San Vicente del Raspeig, Spain

    Juan Manuel González-Forte & Ceneida Fernández

  2. Centre for Instructional Psychology and Technology, University of Leuven, Dekenstraat, 2, 3000, Leuven, Belgium

    Jo Van Hoof & Wim Van Dooren

Authors
  1. Juan Manuel González-Forte
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  2. Ceneida Fernández
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  3. Jo Van Hoof
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  4. Wim Van Dooren
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Correspondence to Juan Manuel González-Forte.

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Juan Manuel González-Forte. Departamento de Innovación y Formación Didáctica, Facultad de Educación, Universidad de Alicante, Calle Aeroplano, s/n. 03690 San Vicente del Raspeig, Spain. E-mail: juanma.gonzalez@ua.es

Current themes of research:

Mathematical thinking, teaching, and learning.

Most relevant publications in the field of Psychology of Education:

No previous publications

Ceneida Fernández. Departamento de Innovación y Formación Didáctica, Facultad de Educación, Universidad de Alicante, Calle Aeroplano, s/n. 03690, San Vicente del Raspeig, Spain. E-mail: ceneida.fernandez@ua.es

Current themes of research:

Mathematical thinking, teaching and learning, teacher training.

Most relevant publications in the field of Psychology of Education:

- Sánchez-Matamoros, G., Fernández, C., & Llinares, S. (2018). Relationships among prospective secondary mathematics teachers’ skills of attending, interpreting and responding to students’ understanding. Educational Studies in Mathematics, DOI: 10.1007/s10649-018-9855-y.

- Ivars, P., Fernández, C., Llinares, S., & Choy, B.H. (2018). Enhancing noticing: Using a hypothetical learning trajectory to improve pre-service primary teachers’ professional discourse. Eurasia. Journal of Mathematics, Science and Technology Education, 14(11), em1599.

- Jiang, R., Li, X., Fernández, C. & Fu, X. (2016). Students’ performance on missing-value word problems: a cross-national developmental study. European Journal of Psychology of Education, 32(4), 551-570.

- Fernández, C., De Bock, D., Verschaffel, L. & Van dooren, W. (2014). Do students confuse dimensionality and “directionality”? Journal of Mathematical Behavior, 36, 166-176.

Jo Van Hoof. Centre for Instructional Psychology and Technology, University of Leuven, Dekenstraat, 2, 3000, Leuven, Belgium. E-mail address: jo.vanhoof@kuleuven.be

Current themes of research:

Conceptual change approach to mathematics learning. Higher order number sense. The role of inhibition in mathematics reasoning. Learners’ rational number understanding. Intuitions, heuristics, and biases in human reasoning.

Most relevant publications in the field of Psychology of Education:

- Van Hoof, J., Degrande, T., Ceulemans, E., Verschaffel, L., & Van Dooren, W. (2018). Towards a mathematically more correct understanding of rational numbers: A longitudinal study with upper elementary school learners. Learning and Individual Differences, 61, 99-108.

- McMullen, J., Van Hoof, J., Degrande, T., Verschaffel, L., & Van Dooren, W. (2018). Profiles of rational number knowledge in Finnish and Flemish students–A multigroup latent class analysis. Learning and Individual Differences, 66, 70-77.

- Bojorque, G., Torbeyns, J., Van Hoof, J., Van Nijlen, D., & Verschaffel, L. (2018). Effectiveness of the Building Blocks program for enhancing Ecuadorian kindergartners’ numerical competencies. Early Childhood Research Quarterly, 44, 231-241.

- Degrande, T., Van Hoof, J., Verschaffel, L., & Van Dooren, W. (2018). Open word problems: taking the additive or the multiplicative road?. ZDM, 50, 1-12.

- Van Hoof, J., Verschaffel, L., Ghesquière, P., & Van Dooren, W. (2017). The natural number bias and its role in rational number understanding in children with dyscalculia. Delay or deficit? Research in Developmental Disabilities, 71, 181-190.

Wim Van Dooren. Centre for Instructional Psychology and Technology, University of Leuven, Dekenstraat, 2, 3000, Leuven, Belgium. E-mail: wim.vandooren@kuleuven.be

Current themes of research:

Mathematical thinking and problem solving. The role of intuitions and heuristics in mathematics. Conceptual change. The use of multiple representations in teaching and learning.

Most relevant publications in the field of Psychology of Education:

- Van Dooren, W., Lem, S,, De Wortelaer, H., & Verschaffel, L. (2018). Improving realistic word problem solving by using humor. Journal of Mathematical Behavior, 53, 96-104

- Van Hoof, J., Degrande, T., Ceulemans, E., Verschaffel, L., & Van Dooren, W. (2018). Towards a mathematically more correct understanding of rational numbers: A longitudinal study with upper elementary school learners. Learning and Individual Differences, 61, 99-108.

- Lem, S., Onghena, P., Verschaffel, L., & Van Dooren, W. (2017). The power of refutational text: Changing intuitions about the interpretation of box plots. European Journal of Psychology of Education, 32(4), 537-550.

- Van Hoof, J., Verschaffel, L., & Van Dooren, W. (2017). Number sense in the transition from natural to rational numbers. British Journal of Educational Psychology, 87(1), 43-56.

- Van Dooren, W., Lehtinen, E., & Verschaffel, L. (2015). Unraveling the gap between natural and rational numbers. Learning and Instruction, 37, 1-4.

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González-Forte, J.M., Fernández, C., Van Hoof, J. et al. Various ways to determine rational number size: an exploration across primary and secondary education. Eur J Psychol Educ 35, 549–565 (2020). https://doi.org/10.1007/s10212-019-00440-w

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