Crossing the cognitive gap between arithmetic and algebra: Operating on the unknown in the context of equations
 Liora Linchevski,
 Nicolas Herscovics
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Abstract
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 prealgebra course.
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 Title
 Crossing the cognitive gap between arithmetic and algebra: Operating on the unknown in the context of equations
 Journal

Educational Studies in Mathematics
Volume 30, Issue 1 , pp 3965
 Cover Date
 19960101
 DOI
 10.1007/BF00163752
 Print ISSN
 00131954
 Online ISSN
 15730816
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Industry Sectors
 Authors

 Liora Linchevski ^{(1)} ^{(2)}
 Nicolas Herscovics ^{(1)} ^{(2)}
 Author Affiliations

 1. School of Education, Hebrew University, Mount Scopus, Jerusalem, Israel
 2. Department of Mathematics and Statistics, Concordia University, Montreal, Canada