Spontaneous Usage of Different Shortcuts Based on the Commutativity Principle

Based on research on expertise a person can be said to possess integrated conceptual knowledge when she/he is able to spontaneously identify task relevant information in order to solve a problem efficiently. Despite the lack of instruction or explicit cueing, the person should be able to recognize which shortcut strategy can be applied - even when the task context differs from the one in which procedural knowledge about the shortcut was originally acquired. For mental arithmetic, first signs of such adaptive flexibility should develop already in primary school. The current study introduces a paper-and-pencil-based as well as an eyetracking-based approach to unobtrusively measure how students spot and apply (known) shortcut options in mental arithmetic. We investigated the development and the relation of the spontaneous use of two strategies derived from the mathematical concept of commutativity. Children from grade 2 to grade 7 and university students solved three-addends addition problems, which are rarely used in class. Some problems allowed the use of either of two commutativity-based shortcut strategies. Results suggest that from grade three onwards both of the shortcuts were used spontaneously and application of one shortcut correlated positively with application of the other. Rate of spontaneous usage was substantial but smaller than in an instructed variant. Eyetracking data suggested similar fixation patterns for spontaneous an instructed shortcut application. The data are consistent with the development of an integrated concept of the mathematical principle so that it can be spontaneously applied in different contexts and strategies

Addends-compare strategy.

<p>The mean processing times in seconds per problem for booklets allowing for a shortcut with the addends-compare strategy (light grey) and the matched baseline booklets (dark grey). The error bars contain the 95% within-participants confidence interval for the comparison between the two booklets. Asterisks indicate significant comparisons. Bars with dashed lines display the relative benefit on addends-compare booklets relative to baseline booklets. The error bar displays the 95% confidence interval of the comparison with zero benefit.</p

Eyetracking indicator of ten-strategy.

<p>Fixation frequency on first, second, and third addend on ten-strategy and baseline booklets charted for the time-course of the problem-solving episodes. For instance, the graph indicates that more than 35% of the fixations taking place within the first fifth of the time a participant has worked on a problem on a baseline booklet were located at the first digit.</p

Eyetracking indicator of addends-compare strategy.

<p>Mean difference between current line of current problem vs. current line fixated. Negative values indicate fixations on preceding problems while positive values result from fixations on subsequent problems. Like in Figure 3, error bars indicate the 95% within subjects CI.</p

Ten-strategy.

<p>Measures analogous to the ones in Figure 1 are displayed for the ten-strategy.</p

Addends-compare strategy on the level of single problems.

<p>Calculation time per problem type in the eyetracking study. Error bars indicate the 95% within subjects CI according to Masson & Loftus [53] based on the error term of the interaction.</p