Year 6 students' cognitive structures and mechanisms for processing tenths and hundredths

This study explored the cognitive functioning of Year 6 students in the domain of decimal-number numeration, particularly with the intention of: (a) comparing the knowledge structure of proficient and semiproficient students with respect to tenths and hundredths knowledge; (b) constructing frameworks and models which explain the structural knowledge differences of proficient and semiproficient students with respect to tenths and hundredths; and (c) drawing implications for instruction. Forty- five students (12 high proficient, 12 semiproficient, 8 medium proficient, 8 medium semiproficient, 5 low proficient) were identified for semistructured individual interviews (Burns, 1994). The interview was informed by the numeration model and, as a consequence, incorporated tasks relating to position and order, to multiplicativity, and to the unitisation and reunitisation of decimal fractions. The interview results revealed that: (a) knowledge of position and order differentiated between high- performing (high proficient, high semiproficient, medium proficient) and low-performing (medium semiproficient, low proficient) students; and (b) availability and accessibility of multiplicativity tasks were the major factors which differentiated performance amongst the high-performing students. As a result of analyses of students' interview responses and the knowledge subcomponents of the decimal-number taxonomy, structural models that represented the cognitions and connections held by the composite performance categories for position/order, multiplicativity, and unitisation/reunitisation were constructed. From a comparison of the structural models, cumulative models that combined findings for each performance category across position/ order, multiplicativity, and unitisation/reunitisation were constructed. The cumulative models represented the two domains involved in decimal-number numeration understanding, namely, whole numbers and fractions, with multiplicativity represented as the structural knowledge that unifies and integrates the structural knowledge of position/order and unitisation/reunitisation. The models were used to draw implications for instruction in decimal numbers and mathematics generally.