Pathways to linking formal and informal knowledge: A comparison of elementary students constructing mathematical knowledge with ratio or part-whole perspectives of rational numbers

It is widely accepted in the mathematics education literature that students construct formal understandings from more informal ones, and that instruction connecting informal and formal knowledge is viewed as a cornerstone of good teaching practice. Research in young children's arithmetic strongly supports this view (Carpenter, 1992). However for rational number the connections that link these two types of knowledge have not been identified for at least two reasons: (a) research with the most prevalent perspective of rational numbers, part-whole, has not been able to demonstrate strong connections between formal and informal knowledge, and (b) there is an alternate perspective, ratio, which builds on different types of informal knowledge. This study examines how students work from these two perspectives to construct mathematical knowledge by documenting both the processes and outcomes that resulted from two groups of fourth grade students constructing knowledge of rational numbers from parallel curricula emphasizing either a ratio or a part-whole perspective of rational numbers. Seven pairs of students in a ratio condition and six pairs of students in a part whole condition were given eight lessons structured around students' use of two types of informal knowledge (i.e. working with contextualized story problems and applying correct relations to decontextualized quantities) and one type of formal knowledge requiring proper usage of rational number notation. Summative differences were assessed by asking the students to (a) sort cards showing rational number representations, (b) compare partitioned areas, and (c) transfer their knowledge to a novel task. Formative differences were assessed by (a) comparing students' patterns of performance on the formal and informal lesson components, (b) targeting strategies used on key lessons, (c) analyzing informal knowledge reported in students' writings, and (d) examining part-whole students' counting practices over the course of the unit. Results indicated that the groups performed similarly on the area partitioning task, but the ratio group displayed knowledge more similar to a formal domain analysis, and higher rates of transfer overall. Qualitative analyses of the students' problem solving suggest that part-whole students were hindered by the type of mental representation they constructed and their reliance on counting strategies.