Unit fractions in the context of proportionality: supporting students' reasoning about the inverse order relationship

Abstract

We analyze a classroom design experiment, conducted in a fourth grade classroom, that served to explore an instructional path in which the introduction of unit fractions and supporting proportional reasoning coincide. Central to this path is the use of means of support in which the objects that unit fractions quantify are not characterized as equal-sized parts of a whole, but as entities that are always separate from a reference unit. We argue that such a path is crucial for helping students develop deep quantitative understandings of fractions, where fraction quantities are, from the very start, linked to the reciprocal and multiplicative relations that their use implies. We focus on the first part of the design experiment in which we helped the students make sense of a concept that is important for initial fraction learning and proportional reasoning, the inverse order relationship among unit fractions.

Promoting middle school students’ proportional reasoning skills through an ongoing professional development programme for teachers

“…Lesh et al (1988) described the inability to understand part-whole relationships; composite units, for example, those used in rate problems; and measurement-related difficulties as "conceptual stumbling blocks" that are critical to the development of primary school students' proportional reasoning (p. 9). Cortina, Visnovska, and Zuniga (2014) noted similar challenges associated with fraction-related concepts.…”

Análisis de las representaciones de los cálculos operacionales en la solución de un problema de movimiento uniforme

“…Incluso hay evidencias de que gran parte de las personas adquieren deficientemente las habilidades relacionadas con este tipo de pensamiento. Los estudiantes de secundaria en su mayoría aprenden reglas de memoria que los capacitan, en el mejor de los casos, para resolver problemas rutinarios, pero que aplican con dificultad en situaciones prácticas o medianamente novedosas comparadas con aquellas en las que recibieron entrenamiento escolar (Ojose, 2015, Tourniaire, 1986, Hoffer, 1988, Wahyuningrum, & Suryadi, 2017, Cortina, Visnovska, & Zuniga, 2014. Modestou, Elia, Gagatsis & Spanoudis (2008) ponen en evidencia las dificultades de los estudiantes para diferenciar entre variación proporcionalidad y lineal.…”

The relational nature of rational numbers

“…The work of McMullen et al (2013McMullen et al ( , 2014 indicates that a cognitive comparison relational scheme, which is not directly related to whole number knowledge, contributes an additional foundation for rational number learning. The power of using such relational understanding for early formal rational number learning has been demonstrated by Cortina, Visnovska and Zuniga (2014). They showed that the capacity to view the measurement unit and the quantity to be measured as separate entities contributed greatly to learning the inverse ordering relationship for unit fractions.…”