A boundary of the second multiplicative concept: the case of Milo

Students entering sixth grade operate with three different multiplicative concepts that influence their reasoning in many domains important for middle school. For example, students who are operating with the second multiplicative concept (MC2 students) can begin to construct fractions as lengths but do not construct improper fractions as numbers. Students who are operating with the third multiplicative concept (MC3 students) can construct both proper and improper fractions as multiples of unit fractions. This paper is a case study of one seventh grade MC2 student, Milo, who demonstrated the most advanced reasoning of all MC2 students in a large project with 13 MC2 and 9 MC3 students. In working on problems involving fractional relationships between two unknowns, most MC3 students constructed reciprocal reasoning. In contrast, the MC2 students struggled with these problems. Similar to the MC3 students, Milo showed some evidence of reciprocal reasoning, and he used proper fractions as operators on unknowns with a rationale. However, Milo did not construct reciprocal reasoning. We account for his reasoning by showing how he used length meanings for proper fractions and how he coordinated two different two-levels-of-units structures. The study expands the mathematics for MC2 students, showing what learning may be possible for students like Milo, and it suggests a change to the theory of students' multiplicative concepts. Specifically, advanced MC2 students are those who have constructed length meanings for fractions and can coordinate two different two-levels-of-units structures