Crossing the cognitive gap between arithmetic and algebra: Operating on the unknown in the context of equations

The objective of the teaching experiment reported in this article was to overcome the cognitive gap, that is, students' inability to spontaneously operate with or on the unknown. Following an analysis of the cognitive obstacles involved, this paper reports the results of an alternative approach. We designed an individualized teaching experiment which was tested in six case studies. In the first part the students' natural tendency to group singletons in the unknown within the equations was expanded to a process of grouping like terms. In the second part we introduced a reverse process to grouping like terms, that of decomposition of a term into a sum. This process, combined with the cancellation of identical terms, provides a procedure for the solution of first degree equations with the unknown on both sides of the equality sign. The last part of the teaching experiment involved the decomposition of an additive term into a difference. The first two parts proved very successful and the students developed procedures on their own that were more efficient than the initial ones. The results of the third part, however, revealed the limits of this approach. The students experienced difficulties in choosing the required decomposition. It seems that some of these obstacles are rather robust and perhaps should not be dealt with incidentally but should be addressed as part of a pre-algebra course.This research was funded by the Quebec Ministry of Education (Founds FCAR 92-ER-1032).