Multicolumn Subtraction

How do skilled students perform this task?What types of errors are made by learners?random errors or systematic errors?factual (arithmetic) errors or procedural errors?incorrect sub-skills or failure to apply sub-skills? Young and O'Shea (1981) hypothesized that many errors are caused by failing to apply a sub-skill.

Children frequently make errors when learning the cognitive skill of multicolumn subtraction. Some of these errors appear to result from number-fact errors - faulty knowledge of basic arithmetic - such as thinking that 7 - 5 = 3. However, many errors appear not to be of this form. There are strong regularities that hold over these other errors. These regularities may be understood as reflecting characteristic faults in the application of the procedure or algorithm for multicolumn subtraction. 

Young and O'Shea (1981) analysed a corpus of 1549 subtraction problem attempts obtained from 33 10-year old children. The attempts contained 344 errors, including 127 which could be interpreted as number-fact errors and 178 which could be interpreted as algorithm errors. Young and O'Shea were unable to classify the remaining 39 errors. Young and O'Shea were unable to classify the remaining 39 errors. Further analysis of the algorithm errors suggested that there were just nine error types. The consistency with which individual children produced the different error types led Young and O'Shea to argue that the behaviour of individual children was best understood in terms of "faithfully executing a faulty algorithm, rather than wrongly following a correct one"

In order to demonstrate this, they developed a production system account of children's behaviour on multicolumn subtraction. The production system was able to account for correct multicolumn subtraction with 12 production rules. Each of the nine algorithm errors could be accounted for by removing rules from the production system or by adding extraneous rules to the production system.

Empirical Findings

Diagnosing Student Models