Language is essential for avoiding surface translations: Associations of students’ spontaneous use of meaning-related phrases for explicating structures with conceptual understanding of multiplication

Abstract There has been a consensus that students’ conceptual understanding of mathematical operations (such as multiplication) can be developed through communication about multiple representations. However, learning opportunities have often appeared to be limited to surface translations (in which only obvious similarities such as numbers have been identified in both representations, without articulating the underlying mathematical structure). Qualitative case studies have suggested that conceptual understanding may develop when students explicitly articulate how the representations are related and when meaning-related language (such as “two groups of four”) is offered to support these explanations. So far, however, the associations of surface translations and meaning-related language use with students’ understanding for multiplication have not been shown quantitatively. The current assessment study was aimed at filling this research gap on this relationship. Fifth and sixth graders’ ( n = 414) conceptual understandings of multiplication were tested through items requesting different translation processes between symbolic, graphical, and textual representations. Students’ responses were scored for correctness and coded with respect to the depth of articulated links between representations in each item and their spontaneous use of meaning-related language phrases. The analysis revealed that surface translations occurred frequently and across all item types. Meaning-related phrases (MRP) were rarely used, most often in problem-posing items (translations from symbolic to textual representations). The frequency of surface translations was strongly correlated with lower total test scores (i.e., also other difficulties). In contrast, students who used at least one MRP significantly outperformed their peers in the total test score and all item types. The study provides first, yet salient quantitative evidence for the continued need to overcome surface translations and the high relevance of promoting meaning-related language for explicating structures to foster students’ conceptual understanding.