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Fourth graders’ expression of the general case

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Abstract

This study forms part of a classroom teaching experiment on the development of a group of 25 9–10 years old students’ algebraic thinking. More specifically, it explored their reasoning while solving word problems built around functional relationships to determine how they generalized through questions posed using natural language, drawn figures or the keyword ‘many’. Their written and oral answers to those questions were analyzed qualitatively to determine which approach most effectively supported the expression of their generalization. The results reveal the benefits of posing questions about the general case in different ways while teaching students to use conventional algebraic representations. According to these findings, representing indeterminate quantities with the keyword ‘many’ induces generalization more successfully than representing them with letters. The use of letters prompts students to seek meaning for the letters, either conventionally, as an unknown or variable quantity, or otherwise, as a label or specific values assigned according to their own criteria. Identifying the most effective procedures may help teachers and curriculum designers formulate mathematical tasks that encourage students to express the generality they perceive in particular cases. Determining the communication demands of each approach is likewise highly useful.

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Notes

  1. Variables and indeterminate quantities are used without being distinguished, as synonyms in this paper.

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Acknowledgements

This work has been developed within the projects EDU2016-75771-P and PID2020-113601GB-I00, financed by the Spanish State Research Agency.

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Authors and Affiliations

  1. University of Almería, Almería, Spain

    Cristina Ayala-Altamirano

  2. University of Salamanca, Salamanca, Spain

    Marta Molina

  3. University of California, Los Angeles, USA

    Rebecca Ambrose

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  1. Cristina Ayala-Altamirano
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  2. Marta Molina
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  3. Rebecca Ambrose
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Correspondence to Cristina Ayala-Altamirano.

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Ayala-Altamirano, C., Molina, M. & Ambrose, R. Fourth graders’ expression of the general case. ZDM Mathematics Education 54, 1377–1392 (2022). https://doi.org/10.1007/s11858-022-01398-8

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