Spontaneously spotting and applying shortcuts in arithmetic—a primary school perspective on expertise

Abstract

One crucial feature of expertise is the ability to spontaneously recognize where and when knowledge can be applied to simplify task processing. Mental arithmetic is one domain in which people should start to develop such expert knowledge in primary school by integrating conceptual knowledge about mathematical principles and procedural knowledge about shortcuts. If successful, knowledge integration should lead to transfer between procedurally different shortcuts that are based on the same mathematical principle and therefore likely are both associated to the respective conceptual knowledge. Taking commutativity principle as a model case, we tested this conjecture in two experiments with primary school children. In Experiment 1, we obtained eye tracking data suggesting that students indeed engaged in search processes when confronted with mental arithmetic problems to which a formerly feasible shortcut no longer applied. In Experiment 2, children who were first provided material allowing for one commutativity-based shortcut later profited from material allowing for a different shortcut based on the same principle. This was not the case for a control group, who had first worked on material that allowed for a shortcut not based on commutativity. The results suggest that spontaneous shortcut usage triggers knowledge about different shortcuts based on the same principle. This is in line with the notion of adaptive expertise linking conceptual and procedural knowledge.

Transferring control demands across incidental learning tasks – stronger sequence usage in serial reaction task after shortcut option in letter string checking

“…The learning-plus-decision perspective on strategy change however suggests that providing participants with the opportunity to find and apply a shortcut in a relatively easy task could strengthen shortcut usage in a more difficult task provided later on. Experiments on shortcut usage in arithmetic ( Godau et al, 2014 ) indeed reflect that that offering an easy-to-find shortcut option can increase later shortcut usage. Therefore, the sequential regularity in the SRT (which can be detected and applied rather easily) could foster later shortcut usage in the alphabet verification task, if the tasks would be applied in the reversed order.…”

Children's understanding of the addition/subtraction complement principle

“…() and Godau et al . () investigated children's understanding of the commutativity principle and found that fixation patterns matched the use of shortcut strategies based on this commutativity principle. To the best of our knowledge, such studies have not been performed to investigate other mathematical principles in children.…”

“…Although children's understanding of the commutativity and inversion principle has received much research attention (e.g., Bisanz, Watchorn, Piatt, & Sherman, ; Canobi, Reeve, & Pattison, ; Gaschler, Vaterrodt, Frensch, Eichler, & Haider, ; Gilmore & Bryant, ; Gilmore & Papadatou‐Pastou, ; Gilmore & Spelke, ; Godau, Haider, Hansen, Schubert, & Gaschler, ; Piatt, Matejko, Watchorn, & Bisanz, ; Prather & Alibali, ; Rasmussen, Ho, & Bisanz, ; Robinson & Dubé, ; Siegler & Stern, ), only few studies investigated children's understanding of the other principles, particularly the addition/subtraction complement principle (e.g., Baroody, ; Canobi, , , ; Dowker, ). This is surprising because a good understanding of the addition/subtraction complement principle is assumed to enhance mathematical proficiency, specifically the application of the subtraction by addition strategy (Baroody et al ., ).…”

Psychological perspectives on expertise

“…Similarly, McCormack et al ( 2014 ) found that expert intensive-care physicians directed their attention to more relevant areas of the situation, compared to competent non-experts. In addition, Godau et al ( 2014 ) found that experts in arithmetic problem solving spontaneously used arithmetic shortcuts. Finally, Ellis and Reingold ( 2014 ) examined the Einstellung effect (i.e., where the first idea that comes to mind blocks finding the best solution to a problem) using this methodology and noted its relevance to understanding expert flexibility (see Bilalić and McLeod, 2014 ).…”