Towards the specification of a computer system to facilitate the learning of the partitive quotient fraction construct

A "Specification Document" was developed for the design of a constructivist, hypermedia learning environment (CHLE) to facilitate the knowledge construction of fractions. Specifications for the design of the CHLE were created from an investigation into children's knowledge construction of fractions. This entailed a review of the mathematical structure of fractions to build fraction tasks which were then used to investigate children's understandings of fractions.

The investigation was couched in the constructivist paradigm. Guba and Lincoln's (1989) methodology of constructivist inquiry incorporating the Hermeneutic Dialectic Circle was adopted. Clinical interviews, talk-aloud protocols and non-participant observations operated within the Hermeneutic Dialectic Circle. Findings from the investigation addressed: (i) teaching interventions, (ii) children's strategies for solving the fraction tasks, and (iii) the goodness of the analog objects used in the tasks.

Teaching interventions involved task sequencing and teaching episodes to assist children over impasses or gaps in their knowledge. Task sequencing moved from circular region models to rectangular region models and length models. One to six analog objects (representative of these three models) were shared between two, three, four, five and six people. Task sequencing was unique for each child and contingent upon each child's knowledge constructions of fractions. Findings indicated that children who did not possess sound knowledge of the partitive quotient fraction construct relied heavily upon their knowledge of the part-whole fraction construct to quantify each person's share. In most cases, this knowledge was fractured and incomplete, therefore most teaching episodes focused on partwhole notions.

Children employed four categories of strategies to solve the tasks. These strategies were categorised according to their ability to: (i) generate fair shares, (ii) facilitate the accurate quantification of each of the shares, and (iii) facilitate a conceptual mapping between the concrete activity and the partitive quotient fraction construct. Category 1 strategies met all three criteria. Category 2 strategies met only the first two criteria. Category 3 strategies met only the first criteria and Category 4 strategies met none of the criteria.

The goodness of the analog objects was judged according to three criteria: (i) ecological validity, (ii) abstract ability, and (iii) ease of partitioning. Findings revealed that the icecream bar (rectangular region model) and the licorice strap (length model) were easier to partition than the pizza and apple pie (circular region models) but lacked ecological validity. The pizza and apple pie had ecological validity but were difficult to partition. The cake ("wide" rectangular region model) did not have ease of partitioning because it supported both vertical and horizontal partitioning. The circular region models of the pikelet and the pancake did not satisfy any of the criteria and were not considered suitable analogs for inclusion in the design of the CHLE.

The findings from the teaching interventions and strategies were translated into "Content" specifications, and the findings from the goodness of the analog objects were translated into "Context" specifications. These two sets of specifications informed the Specification Document for the design of the CHLE. Specifications were presented as: (i) scenarios which correlated instructional objectives of the CHLE with "WHAT" the children should do to meet these instructional objectives and "HOW" the system should address the "WHAT", (ii) interface design as it applies to Object Orientated User Interfaces (OOUI's), and (iii) vignettes which are supported with accompanying screens to indicate screen layout, typography, language, graphics, illustrations, and the look, feel and operation of the system. The CHLE was designed to embody realistic contexts in which children construct their own knowledge as a consequence of their experiences with the fraction tasks.