<p>The present research comprises four experiments designed to explore the role of visual and phonological working memory resources in carry operations or intermediate solutions in complex mental addition and multiplication. A special consideration was given to the effect of arithmetic operation on the relative involvement of visual and phonological resources in complex addition and multiplication. A pilot study was conducted prior to the experiments, aiming to examine the suitability of visual and phonological stimuli for change detection and working memory capacity estimation. Two staff of Victoria University of Wellington with normal or corrected vision attended the pilot study as participants. Pilot Experiments 1 to 4 tested the suitability for probing visual working memory (VWM) capacity of two types of visual stimulus with different feature dimensions: bars of different orientations and Gabor patches with different orientations and spatial frequencies. A single-probe change-detection experimental paradigm was used, with participants making decisions about whether or not probe items were the same as memory items presented previously. Both presentation durations and set sizes were manipulated. Stable estimates of visual working memory capacities were found when Gabor patches with varied spatial frequencies were used, suggesting its utility as a probe for estimating visual working memory capacity. Pilot Experiment 5 was designed to examine the suitability of pronounceable consonant-vowel-consonant non-words as a probe of phonological working memory (PWM). Valid estimates of PWM capacity were found for both participants, suggesting the suitability of phonological non-words as phonological stimuli of assessing PWM capacities and interfering with information phonologically-represented and maintained in working memory. Experiments 1 to 4 investigated the relative involvement of visual and phonological working memory resources in carry operations or intermediate solutions in mental addition and multiplication. Fifty-six undergraduate students of Victoria University of Wellington participated all experiments, and 48 of them provided valid data for final analysis. A dual-task interference paradigm was used in all experiments, with arithmetic tasks and visual/phonological change-detection tasks either performed alone, or simultaneously. For arithmetic tasks, double-digit addition problems and multiplication problems comprising one single-digit and one double-digit were presented horizontally and continuously, and participants reported the final solutions verbally. For visual change-detection tasks, study items were visually presented to participants for 1,000ms before they disappeared. After a 4000ms retention interval, a probe item was presented and participants judged whether the probe item was the same as one of the memory items. For phonological change-detection tasks, phonological nonwords were verbally presented to participants sequentially. After a 4000ms retention interval, a probe nonword was presented to participants, and they indicated whether or not the probe was the same as one of the study non-words. Both numbers of carry operations involved in the arithmetic problems (zero, one, and two) and levels of visual/phonological loads (low, medium, and high) were manipulated in all experiments. For all experiments, the effect of the number of carry operations on calculation performance was observed: arithmetic problems involving more carry operations were solved less rapidly and accurately. This effect was enlarged by concurrent visual and phonological loads, evidenced by significant interactions between task conditions and number of carry operations observed in the accuracy analyses of the arithmetic tasks in all experiments except Experiment 2, in which multiplication problems were solved under visual loads. These findings suggest that both visual and phonological resources are required for the temporary storage of intermediate solutions or carry information in mental addition, while for mental multiplication, only evidence for a role of phonological representations in carry operations was found. For all experiments, the greater performance impairment of carry problems than no-carry problems associated with the presence of working memory loads was not further increased by increasing load level: There were no significant three-way interactions between task conditions, number of carry operations and load levels in accuracy analyses of arithmetic tasks. One possible explanation for this absence of significant three-way interactions might be attributable to some participants switching between phonological and visual working memory for the temporary storage of carrier information or intermediate solutions as a result of decreasing amount of available phonological or visual working memory resources. In conclusion, the findings of the present research provide support for a role of both visual and phonological working memory resources in carry operations in mental addition, and a role of phonological working memory resources in carry operation in mental multiplication. Thus, it can be concluded that solving mental arithmetic problems involving carry-operations requires working memory resources. However, these results contradict the prediction of the Triple Code Model, which assumes addition mainly relies on visual processing, and multiplication mainly relies on verbal processing, while complex mental arithmetic is solved with the aid of visual processing regardless of the arithmetic operation. Thus, these results challenge the operation-specific involvement of working memory resources in complex mental arithmetic. However, it should be noted that the same arithmetic problems were solved three times by the same participants, which might have encouraged more activation in phonological processing than visual processing due to the practice effect.</p>
Fast Two-Stage Computation of an Index Policy for Multi-Armed Bandits with Setup Delays
