Prospective Teachers’ Diagnostic Thinking on Students’ Understanding of Multi-Digit Multiplication: A Content-Related Analysis on Unpacking of Knowledge Elements

<jats:title>Abstract</jats:title><jats:p>Teachers’ in-depth diagnostic thinking has been shown to be crucial for student-centered teaching as they need to perceive and interpret students’ understanding for well-informed decision-making on adaptive teaching practices. The paper presents a content-related approach to analyzing diagnostic thinking processes with respect to the mathematical knowledge elements that prospective teachers identify as students’ resources and obstacles. Prospective teachers’ challenge is that some relevant knowledge elements first have to be unpacked, because compact concepts (such as the place value concept) or procedures (such as for multi-digit multiplication) comprise several smaller knowledge elements (such as the positional property) that have to be made explicit for students to foster their learning processes adequately. Our study examines what knowledge elements prospective teachers perceive and interpret in a transcript vignettes on multi-digit multiplication (of decimal and natural numbers) and its underlying basic arithmetic concepts (place value understanding and meaning of multiplication) in written diagnostic judgments on students’ resources and obstacles (<jats:italic>N</jats:italic> = 196). A comparative design within the vignette is used to investigate how far the process of perceiving can be supported by thematic cues. The analysis reveals that those knowledge elements cued in the vignette by being already unpacked and explicitly addressed are perceived and interpreted more often (but with lower correctness) than those that are uncued and therefore have to be unpacked by the prospective teachers themselves. This confirms the need to prepare prospective teachers for unpacking mathematical concepts themselves.</jats:p>