An examination of third- and fourth-graders’ equivalence knowledge after classroom instruction

The present correlational study examined third- and fourth-graders’ (N = 56) knowledge of mathematical equivalence after classroom instruction on the equal sign. Three distinct learning trajectories of student equivalence knowledge were compared: those who did not learn from instruction (Never Solvers), those whose performance improved after instruction (Learners), and those who had strong performance before instruction and maintained it throughout the study (Solvers). Learners and Solvers performed similarly on measures of equivalence knowledge after instruction. Both groups demonstrated high retention rates and defined the equal sign relationally, regardless of whether they had learned how to solve equivalence problems before or during instruction. Never Solvers had relatively weak arithmetical (nonsymbolic) equivalence knowledge and provided operational definitions of the equal sign after instruction.

Improving Understanding of Mathematical Equivalence

Modify arithmetic problem formats to make the relational equation structure more transparent. We describe this practice and three additional evidence-based practices: (1) introducing the equal sign outside of arithmetic, (2) concreteness fading activities, and (3) comparing and explaining different problem formats and problem-solving strategies.

Solar System tests in Brans–Dicke and Palatini $$f({\mathcal {R}})$$-theories

We compare Mercury’s precession test in standard general relativity, Brans–Dicke theories (BD), and Palatini $$f({\mathcal {R}})$$ f ( R ) -theories. We avoid post-Newtonian approximation and compute exact precession in these theories. We show that the well-known mathematical equivalence between Palatini $$f({\mathcal {R}})$$ f ( R ) -theories and a specific subset of BD theories does not extend to a really physical equivalence among theories since equivalent models still allow a different incompatible precession for Mercury depending on the solution one chooses. As a result one cannot use BD equivalence to rule out Palatini $$f({\mathcal {R}})$$ f ( R ) -theories. On the contrary, we directly discuss that Palatini $$f({\mathcal {R}})$$ f ( R ) -theories can (and specific models do) easily pass Solar System tests as Mercury’s precession.

Young students’ understanding of mathematical equivalence across different schools in South Africa

Background: Mathematical equivalence is a critical element of arithmetic understanding and a key component of algebraic thinking which is necessary for success in all levels of mathematics. Research studies continue to highlight misconceptions related to equivalence and reveal that many primary school students have a narrow and limiting view of the equals sign as an operation.Aim: This study aims to investigate young students’ understanding of mathematical equivalence in South Africa with a particular focus on their interpretations of the equals sign.Setting: Research data was obtained from students across six schools from different contexts within the Western Cape.Methods: We gave students an adapted standardised assessment containing 15 items related to equivalence.Results: Our analyses indicated that students focus more on the equals sign as an operation which involves calculating an answer. While some referred to equivalence as meaning the same as, most of them were inclined to accept the operational definition of the equals sign (i.e. the answer to the problem) as a better and preferred definition. In addition, student performance was poor on equation-solving problems and they rarely used comparative relational strategies in their solutions.Conclusion: The findings of this research confirmed that difficulties with equivalence reported by earlier research is widespread across this group of grade 4 students. This has implications for both curriculum, textbook and materials design and teacher professional development.