Middle school students’ construction of reciprocal reasoning with unknowns

Hackenberg A. J. , SEVİNÇ Ş.

Journal of Mathematical Behavior, vol.65, 2022 (ESCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 65
  • Publication Date: 2022
  • Doi Number: 10.1016/j.jmathb.2021.100929
  • Journal Name: Journal of Mathematical Behavior
  • Journal Indexes: Emerging Sources Citation Index (ESCI), Scopus, Academic Search Premier, EBSCO Education Source, Educational research abstracts (ERA), Psycinfo
  • Keywords: Iterative fraction scheme, Rational number knowledge, Reciprocal reasoning, Algebraic reasoning, Reversible reasoning, Unknown, TEACHERS
  • Middle East Technical University Affiliated: Yes


© 2021 Elsevier Inc.Three iterative, after school design experiments with small groups of middle school students were conducted to investigate how students represent fractional relationships between two unknowns and whether they construct reciprocal reasoning with unknowns. Of the 22 students who participated, 9 had constructed fractions as multiples of unit fractions. Seven of these 9 students constructed reciprocal reasoning with unknowns. They did so by constructing a multiplicative relationship between a unit fraction of one unknown and the other unknown. The other 2 students did not make this construction. Instead, they represented relationships between unknowns with whole number multiplication and division, and by adding unknowns and fractional parts of unknowns. The students who constructed reciprocal reasoning demonstrated a link between fractions as measures and fractions as operators, one benefit of working on rational number knowledge and algebraic reasoning together. Implications for teaching include recommendations for supporting students to construct reciprocal reasoning with unknowns.