Affordances from Number Lines in Fractions Instruction: Students’ Interpretation of Teacher’s Intentions

Abstract

Given its pedagogical appeal, the number line is a commonly used representation in the teaching and learning of fractions. However, behind its apparent simplicity, this mathematical object may involve layers of complexity when looked at from the perspective of affordances as is the case in this study. In particular, this in situ exploration examined the affordances of the number line as a mathematical object in fractions instruction. Watson’s (Res Math Educ 9(1), 111–126, 2007) analytical framework was used to scrutinise moment-by-moment teaching sequences from videotaped data collected from four 7th grade lessons conducted by 2 teachers. The results show the dissonance that may arise between teachers’ intentions and the accompanying students’ interpretations when the number line is the object of discussion. The teachers inadvertently used the area and measurement model of fractions almost simultaneously with the result that students were constrained to make sense of the teachers’ intention. Consequently, the 2 teachers made several instructional attempts to enable the affordances concealed in the number line to become more perceptible. Such instructional attempts took the form of modified tasks, questions, prompts and actions that the teachers brought to the fore spontaneously. Aligning with known difficulties associated with number lines, this study further highlights the instructional subtleties that may be necessary in making affordances visible to help students understand the measure meaning of fractions. Explicitness in instruction is critical to enable the affordances inherent in a mathematical object to be perceptible to allow access to concepts.