Exploring the Learning of Ratio by a Semiotic Process in a Combined Fifth- and Sixth-Grade Class

Published studies support the premise that proportional reasoning, and specifically, the concepts of rate and ratio, contribute to the difficulty students face in mathematics learning and performance. Learning rate or ratio appears to require the ability to recognize and apply strategies pertinent to one of several subconstructs of proportional reasoning. There is an apparent difficulty students have in transferring strategy and technique among situations requiring application of a particular subconstruct. This implies a discomfort in properly assigning meaning to a problem situation in an attempt to apply the appropriate subconstruct. Early 20th-century researchers, notably de Saussure and Vygotsky, explored a process of signification whereby symbols, words, and tools acquire meanings that incorporate a set of prior signs, deriving from other processes of signification or from basic elements of language and culture. Previous studies have identified a process of signification in basic concepts such as place value in early elementary grades. This study attempted to further identify, through interviews of a 10-student sample, a process of signification among high-aptitude fifth- and sixth-grade students that led to functional understanding of the concept of rate and ratio. Such a process was identified if the 10 students were taken collectively. For solving problems involving proportional reasoning, the students preferred solution methods that were fully signified semiotically, or that led most directly to the solution. Full signification of a concept promoted a student's perception that he or she "owned" the concept