Abstract:

Mental arithmetic research has been hindered, in part, by indirect methods of investigation. For example, some researchers have defined mental flexibility as the choice of the most appropriate solution to a problem (1, 2). Others have examined computational accuracy and timeliness (3). What is unclear, however, is how appropriateness (strategic “fit”), accuracy, or speed can inform our understanding of the mental strategies that underlie these outcomes.

This paper reports on a novel methodology developed to investigate directly cognitive flexibility in mental arithmetic (4). The method was tested with 43 elementary students:
(1) to determine its suitability for differentiating between rigid and flexible mental strategies,
(2) to assess the extent to which it generates a variety of flexible mental strategies, and
(3) to identify the range and exemplars of rigid and flexible strategies exhibited.

Twelve two-digit addition and subtraction problems were displayed on individual 3 x 5” cards (33+33; 34+36; 47+28; 56+29; 65+35; 73+26; 31-29; 46-19; 63-25; 66-33; 88-34; 95-15). Problems were designed to represent one or more numerical patterns or relationships: double digits, double facts, inverse problem, factors close to ten, double fives at the ones place, sums of ten at the ones place, small range, and regrouping.

Students were asked to examine each card and to sort them individually into “easy” or “hard” categories. Brief interviews elicited student reasoning underlying the sorting decisions. Students were also asked to compare their reasoning for pairs of related problems (e.g., 33+33 and 66-33).

Results include the ranking of problems from easiest to hardest and a comparison of characteristics for problems sorted as “easy” versus “hard.” Examples of cognitive rigidity and cognitive flexibility are reported. The results are discussed in terms of the three research questions. It is argued that the methodology of this study overcomes limitations of earlier studies and opens up new possibilities for research about mental arithmetic that could not previously have been studied.

References:
[1] Star, J.R., & Newton, K.J. (2009). The nature and development of experts’ strategy flexibility for solving equations. ZDM - The
International Journal on Mathematics Education, 41(5), 557-567.
[2] Verschaffel, L., Luwel, K., Torbeyns, J., & Van Dooren, W. (2009). Conceptualizing, investigating, and enhancing adaptive expertise in elementary mathematics education. European Journal of Psychology of Education, 24(3), 335-359.
[3] Torbeyns, J., De Smedt, B., Ghesquière, P., & Verschaffel, L. (2009). Jump or compensate? Strategy flexibility in the number domain up to 100. ZDM - The International Journal on Mathematics Education, 41(5), 581-590.
[4] Rathgeb-Schnierer, E., & Green, M. (2015). Cognitive flexibility and reasoning patterns in American and German elementary students when sorting addition and subtraction problems. Paper presented at the 9th Congress of European Society for Research in Mathematics Education, Prague, Czech Republic, February 4 – 8.

Keywords:

Mental arithmetic, cognitive flexibility, cognitive rigidity.