Rational Number Learning in Linear Measurement

It has been suggested that children need both a deeper and more overarching sense of how
the various representations of rational number relate to each other. The research that I will
present is based on collaborations with psychologist Robbie Case in which we developed
an experimental curriculum that fosters an understanding of these interrelationships. Our
proposal is that a core conceptual organizing structure for rational number understanding
is formed by the coordination of children's intuitive understanding of proportion and their
numerical splitting schemas. In order to support this coordination the curriculum we
developed introduces rational number through the teaching of percent in a linear
measurement context. This approach highlights rational number as operator and measure
rather than focusing on the conventional part-whole introduction to this number system . In
our curriculum we alter the traditional sequence of instruction so that fractions and
decimals are taught later -- and are grounded in students’ acquired knowledge of percents.
Initial props, such as drainage pipes with moveable venting tubes and cylindrical beakers,
allow students to make ratio and proportional judgments of the fullness of these containers
relative to the whole using the language of percents. So far we have implemented this
curriculum in 8 classrooms and our results revealed that students showed flexible
movement among representations, resistance to misleading cues, and the ability to order
numbers by magnitude. In this presentation I describe the curriculum and the measures
that we designed, and present qualitative and quantitative analyses of the results of the
studies.

Related articles and chapters

Moss, J. (2005). Pipes, tubes, and beakers: Teaching rational number. In J. Bransford
and S. Donovan (Eds.), How Children Learn: History Science and Mathematics in the
Classroom (pp. 309-351). Washington, D.C.: National Academy Press.

Moss, J. (2002). Percents and proportion at the center: Altering the teaching sequence
for rational number. In B. Littweiller (Ed.), Making Sense of Fractions , Ratios, and
Proportions (pp. 109-120). Reston, VA: National Council of Teachers of Mathematics.

Moss, J. (2003, Feb.). On the way to computational fluency: Beginning with percents as a
way of developing understanding of the operations in rational numbers. Teaching
Children Mathematics, 9(6), 335-346.

Kalchman, M., Moss, J., & Case, R. (2001). Psychological models for the development of
mathematical understanding : Rational numbers and functions. In S. Carver & D. Klahr
(Eds.) Cognition and Instruction: 25 years of progress (pp. 1-39). Mahweh, NJ: Lawrence
Erlbaum Associates, Inc..

Moss, J. & Case, R. (1999). Developing children’s understanding of the rational numbers: A new
model and an experimental curriculum. Journal for Research in Mathematics Education. 30(2),
122-147.