Acquisition and use of shortcut strategies by traditionally schooled children

Abstract

This study aimed at analysing traditionally taught children's acquisition and use of shortcut strategies in the number domain 20-100. One-hundred-ninety-five second, third, and fourth graders of different mathematical achievement levels participated in the study. They were administered two tasks, both consisting of a series of two-digit additions and subtractions that maximally elicit the use of the compensation 45 þ 29 ¼ ; 45 þ 30 À 1 ¼ 75 À 1 ¼ 74 ð Þ and indirect addition strategy (71 À 68 ¼ ; 68 þ 2 ¼ 70; 70 þ 1 ¼ 71, so the answer is 2+1 or 3). In the first task, children were instructed to solve all items as accurately and as fast as possible with their preferred strategy. The second task was to generate at least two different strategies for each item. Results demonstrated that children of all grades and all achievement levels hardly applied the compensation and indirect addition strategy in the first task. Children's strategy reports in the second task revealed that younger and lower achieving children did not apply these strategies because they did not (yet) discover these strategies. By contrast, older and higher achieving children appeared to have acquired these strategies by themselves. Results are interpreted in relation to cognitive psychological and socio-cultural perspectives on children's mathematics learning.

Special education students’ use of indirect addition in solving subtraction problems up to 100—A proof of the didactical potential of an ignored procedure

“…First of all, student characteristics, such as their general mathematical ability, age, and grade level (see Torbeyns et al, 2009b), are found to be of influence. Furthermore, teaching characteristics, for example, whether or not students have been taught a particular procedure, turned out to play a role, although not all researchers found this (see Section 1.3).…”

“…The student's use of materials and models is left out as well b These calculation steps are not very common to solve this problem; they are only given to explain this particular combination of procedure and strategy students are able to solve subtraction problems by applying IA. For example, a few recent intervention studies concluded that even students in regular primary education have difficulties using IA to solve subtraction problems (Torbeyns et al, 2009a;Torbeyns, De Smedt, Ghesquière, & Verschaffel, 2009b). Torbeyns et al (2009a) found that students have great difficulty picking up the IA procedure, even when they have been taught IA.…”

“…To contribute to this broad understanding of subtraction, students should be given more than just bare number problems. Several studies (Klein et al, 1998;Torbeyns et al, 2009b;Blöte et al, 2001;Van den Heuvel-Panhuizen, 1996) revealed that bare number problems hardly evoke the use of IA, which can be explained by the presence of the minus sign that emphasizes the "taking-away" action (Van den Heuvel-Panhuizen, 1996). Context problems, on the contrary, lack this operation symbol and therefore open up both interpretations of subtraction (Van den Heuvel-Panhuizen, 2005).…”

Sum and difference compensation strategy as a calculus strategy

“…У нашој образовној пракси правила стал-ности збира и разлике се до четвртог разреда основне школе системски изучавају, и то кроз сва три вида изражавања, док се у страној лите-ратури (Carpenter et al, 1998;Fuson et al, 1997;Torbeyns et al, 2009) правила спомињу и користе као стратегија у рачунању, без експлицитног из-двајања и представљања овог правила. Правило називају стратегијом компензације.…”

“…Међутим, када им је страте-гија објашњена на хипотетичком задатку, знат-но већи број ученика је користио наведену стра-тегију као свој начин при рачунању. У истражи-вању које се бавило испитивањем неформалних стратегија рачунања (Torbeyns et al, 2009) по-себна пажња посвећена је стратегији компенза-ције. Аутори су истраживали да ли заиста -како је једно друго истраживање показало (Shrager & Siegler, 1998) -ученици могу спонтано да развију стратегију компензације при рачунању, иако то није било предмет систематске обраде.…”