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The development of students’ use of additive and proportional methods along primary and secondary school

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Abstract

This study investigates the development of proportional and additive methods along primary and secondary school. In particular, it simultaneously investigates the use of additive methods in proportional word problems and the use of proportional methods in additive word problems. We have also studied the role played by integer and non-integer relationships between the given numbers and the nature of quantities (discrete or continuous) in the development of these phenomena. A test consisting of additive and proportional missing-value word problems was solved by 755 primary and secondary school students (from fourth to tenth grade). The findings indicate that the use of additive methods in proportional situations increased during primary school and decreased during secondary school, whereas the use of proportional methods in additive situations increased along primary and secondary school. Moreover, the presence or absence of integer ratios strongly affects this behavior, but the nature of quantities only has a small influence on the use of proportional methods.

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Notes

  1. Details about the percentages and p values related to these pairwise comparisons for proportional answers can be found in Appendix 1.

  2. Details about the percentages and p values related to these pairwise comparisons for additive answers can be found in Appendix 2.

  3. Details about the percentages and p values related to these pairwise comparisons for proportional methods can be found in Appendix 3.

  4. Details about the percentages and p values related to these pairwise comparisons for additive methods can be found in Appendix 3.

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Author information

Authors and Affiliations

  1. Departamento de Innovación y Formación Didáctica, Facultad de Educación, Universidad de Alicante, Campus de San Vicente del Raspeig, 03080, Alicante, Spain

    Ceneida Fernández & Salvador Llinares

  2. Center for Instructional Psychology and Technology, Katholieke Universiteit Leuven, Vesaliusstraat 2 PB 3770, 3000, Leuven, Belgium

    Wim Van Dooren, Dirk De Bock & Lieven Verschaffel

  3. Hogeschool-Universiteit Brussel, Brussels, Belgium

    Dirk De Bock

Authors
  1. Ceneida Fernández
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  2. Salvador Llinares
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  3. Wim Van Dooren
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  4. Dirk De Bock
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  5. Lieven Verschaffel
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Corresponding author

Correspondence to Ceneida Fernández.

Additional information

Ceneida Fernández. Departamento de Innovación y Formación Didáctica, Facultad de Educación, Universidad de Alicante, Campus de San Vicente del Raspeig, 03080 Alicante, Spain. E-mail: ceneida.fernandez@ua.es

Current themes of research:

Mathematics knowledge acquisition. Mathematics education. Computer-supported learning.

Relevant publications:

Fernandez, C., Llinares, S., Van Dooren, W., De Bock, D. & Verschaffel, L. (2011). Effect of number structure and nature of quantities on secondary school students’ proportional reasoning. Studia Psychologica, 53(1), 69–81.

Fernández, C. & Llinares, S. (2011). From the additive to the multiplicative structure: The Effect of two variables in the development of proportional reasoning. Infancia & Aprendizaje, 34(1), 67–80.

Llinares, S. & Roig, A. (2008). Secundary school students’ construction and use of mathematical models in solving word problems. International Journal of Science and Mathematics Education, 6(3), 505–532.

Salvador Llinares. Departamento de Innovación y Formación Didáctica, Facultad de Educación, Universidad de Alicante, Campus de San Vicente del Raspeig, 03080 Alicante, Spain. E-mail: sllinares@ua.es

Current themes of research:

Mathematics knowledge acquisition. Mathematics education. Computer-supported learning.

Relevant publications:

Fernandez, C., Llinares, S., Van Dooren, W., De Bock, D. & Verschaffel, L. (2011). Effect of number structure and nature of quantities on secondary school students’ proportional reasoning. Studia Psychologica, 53(1), 69–81.

Fernández, C. & Llinares, S. (2011). From the additive to the multiplicative structure: The Effect of two variables in the development of proportional reasoning. Infancia & Aprendizaje, 34(1), 67–80.

Llinares, S. & Roig, A. (2008). Secundary school students’ construction and use of mathematical models in solving word problems. International Journal of Science and Mathematics Education, 6(3), 505–532.

Wim Van Dooren. Center for Instructional Psychology and Technology, Katholieke Universiteit Leuven, Vesaliusstraat 2 PB 3770, 3000 Leuven, Belgium. E-mail: wim.vandooren@ped.kuleuven.be

Current themes of research:

Problem solving in mathematics in general and in geometry, algebra and probability in particular. The use of representations in mathematical reasoning and problem solving. Intuitions, heuristics and biases in human reasoning. The conceptual change approach to mathematics and science teaching and learning Adaptive and routine expertise in mathematics.

Relevant publications:

Van Dooren, W., De Bock, D., Vleugels, K. & Verschaffel, L. (2010). Just answering or thinking? Contrasting pupils’ solutions and classifications of missing-value word problems. Mathematical Thinking and Learning, 12(1), 20–35.

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.

Van Dooren, W., De Bock, D., Evers, M. & Verschaffel, L. (2009). Students’ overuse of proportionality on missing-value problems: How numbers may change solutions. Journal for Research in Mathematics Education, 40(2), 187–211.

Dirk De Bock. Center for Instructional Psychology and Technology, Katholieke Universiteit Leuven, Vesaliusstraat 2 PB 3770, 3000 Leuven, Belgium. E-mail: dirk.debock@ped.kuleuven.be

Current themes of research:

Problem solving in mathematics. The use of representations in mathematical reasoning and problem solving. Adaptive and routine expertise in mathematics.

Relevant publications:

De Bock, D., Verschaffel, L., Van Dooren, W., Deprez, J. & Roelens, M. (2011). Abstract or concrete examples in learning mathematics? A replication and elaboration of Kaminski, Sloutsky, and Heckler’s study. Journal for Research in Mathematics Education, 42(2), 109–126.

De Bock, D., Van Dooren, W. & Verschaffel, L. (2011). Students’ over-use of linearity: An exploration in physics. Research in Science Education, 41(3), 389–412.

Lieven Verschaffel. Center for Instructional Psychology and Technology, Katholieke Universiteit Leuven, Vesaliusstraat 2 PB 3770, 3000 Leuven, Belgium. E-mail: lieven.verschaffel@ped.kuleuven.be

Current themes of research:

Teaching and learning of problem-solving and learning skills. Conceptual change. Metacognitive and affective aspects of learning. Computer-supported learning. Mathematics education.

Relevant publications:

Verschaffel, L., Reybrouck, M., Janssens, M. & Van Dooren, W. (2010). Using graphical notations to assess children’s experiencing of simple and complex musical fragments. Psychology of Music, 38(3), 259–284.

Verschaffel, L., Reybrouck, M., Jans, C. & Van Dooren, W. (2010). Children’s criteria for representational adequacy in the perception of simple sonic stimuli. Cognition and Instruction, 28(4), 475–502.

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(3), 335–359.

Appendices

Appendix 1

Table 4 Pairwise comparations (p-values). Percentages of proportional answers (Prop) on proportional and additive problems
Full size table

Appendix 2

Table 5 Pairwise comparations (p values). Percentages of additive answers (Add) on proportional and additive problems
Full size table

Appendix 3

Table 6 Pairwise comparations (p values). Percentages of proportional answers (Prop) and additive methods (Add) on integer and non-integer versions
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Fernández, C., Llinares, S., Van Dooren, W. et al. The development of students’ use of additive and proportional methods along primary and secondary school. Eur J Psychol Educ 27, 421–438 (2012). https://doi.org/10.1007/s10212-011-0087-0

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  • DOI: https://doi.org/10.1007/s10212-011-0087-0

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