Elementary Subtraction Research Articles

Use of indirect addition in adults’ mental subtraction in the number domain up to 1,000

This study examined adults' use of indirect addition and direct subtraction strategies on multi-digit subtractions in the number domain up to 1,000. Seventy students who differed in their level of arithmetic ability solved multi-digit subtractions in one choice and two no-choice conditions. Against the background of recent findings in elementary subtraction, we manipulated the size of the subtrahend compared to the difference and only selected items with large distances between these two integers. Results revealed that adults frequently and efficiently apply indirect addition on multi-digit subtractions, yet adults with higher arithmetic ability performed more efficiently than those with lower arithmetic ability. In both groups, indirect addition was more efficient than direct subtraction both on subtractions with a subtrahend much larger than the difference (e.g., 713 - 695) and on subtractions with a subtrahend much smaller than the difference (e.g., 613 - 67). Unexpectedly, only adults with lower arithmetic ability fitted their strategy choices to their individual strategy performance skills. Results are interpreted in terms of mathematical and cognitive perspectives on strategy efficiency and adaptiveness.

Inverse reference in adults-elementary arithmetic.

Mauro, LeFevre, and Morris (2003) and Campbell (2008) manipulated problem format to assess university students' simple division and subtraction. Large division problems (dividend > 25; e.g., 42 / 6 = _) and large subtraction problems (minuend > 10; e.g., 13 - 6 = _), but not small problems, were solved more quickly when presented in inverse operation format (e.g., 6 x _ = 42 for division; 6 + _ = 13 for subtraction). They concluded that adults often solve large simple division and subtraction problems by reference to the inverse operation but rely on direct memory retrieval for smaller problems. Their findings, however, might have resulted from unequal practice or mixing of the inverse operations. Here, in Experiment 1 (division) and Experiment 2 (subtraction) normal and inverse formats received equal practice and only one operation was practiced (i.e., division or subtraction). Large divisions and subtractions were solved substantially faster when presented in inverse format, but there was also evidence that subtraction ties (e.g., 12 - 6 = 6) and small subtractions (minuend <or=10) benefited from inverse format. The results affirm that inverse reference is an important element in adult's performance of elementary subtraction and division.

Elementary subtraction.

Four experiments examined performance on the 100 "basic facts" of subtraction and found a discontinuous "stair step" function for reaction times and errors beginning with 11 - n facts. Participants' immediate retrospective reports of nonretrieval showed the same pattern in Experiment 3. The degree to which elementary subtraction depends on working memory (WM) was examined in a dual-task paradigm in Experiment 4. The reconstructive processing used with larger basic facts was strongly associated with greater WM disruption, as evidenced by errors in the secondary task: this was especially the case for participants with lower WM spans. The results support the R. S. Siegler and E. Jenkins (1989) distribution of associations model, although discriminating among the alternative solution processes appears to be a serious challenge.

Feature Based Modelling: A methodology for producing coherent, consistent, dynamically changing models of agents' competencies

Feature Based Modelling uses attribute value machine learning techniques to model an agent's competency. This is achieved by creating a model describing the relationships between the features of the agent's actions and of the contexts in which those actions are performed. This paper describes techniques that have been developed for creating these models and for extracting key information therefrom. An overview is provided of previous studies that have evaluated the application of Feature Based Modelling in a number of educational contexts including piano keyboard playing, the unification of Prolog terms and elementary subtraction. These studies have demonstrated that the approach is applicable to a wide spectrum of domains. Classroom use has demonstrated the low computational overheads of the technique. A new study of the application of the approach to modelling elementary subtraction skills is presented. The approach demonstrates accuracy in excess of 90% when predicting student solutions. It also demonstrates the ability to identify and model student's buggy arithmetic procedures.