The design of a computer-based pedagogy for teaching calculator representations

This thesis demonstrates a novel method for improving the understanding of numerical structure in arithmetic through the use of computer-based multiple linked external representations. The system ENCAL exploits three representations: iconic, calculator and dataflow. Small-scale studies contributed to the design, and the results of a final evaluation study suggest that the approach can be usefully exploited in classroom mathematics education. Cognitive science research has extended the concept of internal mental structures to include the interactions which take place between a person and the environment, such as technology-supported learning environments (Kozma, et al., 1996). The following thesis asserts that a computer-based learning environment facilitates the construction and use of mental models, particularly if one advocates the idea that cognition is viewed not as a purely mental process, but as a system which includes the individual, his/her social context, and the available cognitive tools - such as a computer (Dörfler, 1993). In addition, computer-based learning environments aid concrete to abstract thinking,
because visually concrete objects can be linked to more formal and abstract mathematical representations (Kaput, 1989).
The mathematical problem solving ability of school children aged 12-13 years was assessed using the computer-based learning program ENCAL. The system helps children develop their concept of number and their skills with multiplication and addition with the help of a software calculator and some additional computer-based support. The aim of the evaluation was to ascertain the effectiveness of the software’s three equivalent and linked representations: iconic(concrete); datatree (intermediate)-, and calculator (abstract)-, with regard to helping pupils solve text-based arithmetic problems. Two groups of mixed ability children were tested, one group had use of the intermediate data tree whereas the other group did not. Overall, it was found that both high and low attainment pupils in the data tree group obtained a greater number of totally correct answers compared to the no data tree group. Also, the low attainment pupils in the data tree group achieved notably more correct calculator answers for the most difficult question. In particular, those participants who did not have access to the data tree had operator and brackets (i.e. parentheses) problems in all three of the questions. However, the group who were able to use the data tree had
no operator errors, and only two brackets errors with the most difficult question.