The influence of theoretical mathematical foundations on teaching and learning: a case study of whole numbers in elementary school

Abstract

This paper examines the existence and impact of theoretical mathematical foundations on the teaching and learning of whole numbers in elementary school in France. It shows that the study of the New Math reform –which was eventually itself replaced in the longer term – provides some keys to understanding the influence of mathematical theories on teaching and learning. The paper studies changes related to place value, a notion that was deeply impacted by the introduction of numeration bases other than ten in 1970, and their subsequent removal in the 1980s. What the author terms ‘numeration units’ (ones, tens, hundreds, thousands, etc.) and ‘powers-of-ten written in figures’ (1, 10, 100, 1000, etc.) are key tools for describing and understanding changes. The author identifies two theories that have formed the basis for place value teaching in the twentieth century, and examines some aspects of their influence. The paper also addresses epistemological issues in the relation between academic mathematics and school mathematics, and highlights the role of units in the teaching of basic arithmetic.

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Notes

  1. 1.

    In France, the New Math elementary school syllabus was published in 1970. This was followed by a counter-reform, with syllabi published between 1977 and 1980 (depending on the class level).

  2. 2.

    The wording is somewhat different: e.g., ‘scholarly knowledge’ corresponds to ‘mathematical theory’, while ‘knowledge to be taught’ corresponds to ‘the planned mathematics curriculum’.

  3. 3.

    The early post-reform period saw an attempt to theorize based on ‘number words’ computations: e.g., 492 = (4 × 100) + (4 × 20) + 12, which reads four (4) hundreds (100), four (4) twenties (20), twelve (12) (in French).

  4. 4.

    Unlike the English language, the French language does not use the number name to form the unit name. However, working in different bases requires using several names for the unit of a given order: e.g., twos, threes, tens in the first order; fours, nines, hundreds, in the second.

  5. 5.

    Other units are taught in the context of continuous quantities such as length, mass, etc. In the context of numbers, the wording is restricted to the number 1 (and the first place in the numeration chart in French).

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Acknowledgements

This article is based on a paper that was presented at the pre-conference for the 23rd ICMI study on Whole Number Arithmetic in Macau (China), in June 2015.

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Correspondence to Christine Chambris.

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Chambris, C. The influence of theoretical mathematical foundations on teaching and learning: a case study of whole numbers in elementary school. Educ Stud Math 97, 185–207 (2018). https://doi.org/10.1007/s10649-017-9790-3

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Keywords

  • New math reform
  • Relation between school and academic mathematics
  • Didactic transposition
  • Place value
  • Numeration unit
  • Unit