Conceptual knowledge of the associativity principle: A review of the literature and an agenda for future research.

Individuals use diverse strategies to solve mathematical problems, which can reflect their knowledge of arithmetic principles and predict mathematical expertise. For example, '6 + 38 - 35' can be solved via '38 - 35 = 3' and then '3 + 6 = 9', which is a shortcut-strategy derived from the associativity principle. The shortcut may be critical for understanding algebra, however approximately 50% of adults fail to use it. We review the research to consider why the associativity principle is challenging and highlight an important distinction between shortcut identification and execution. We also discuss how domain-specific skills and domain-general skills might play an important role in shortcut identification and execution, and provide an agenda for future research.

Investigating the role of attention in the identification of associativity shortcuts using a microgenetic measure of implicit shortcut use.

Many mathematics problems can be solved in different ways or by using different strategies. Good knowledge of arithmetic principles is important for identifying and using strategies that are more sophisticated. For example, the problem "6 + 38 - 35" can be solved through a shortcut strategy where the subtraction "38 - 35 = 3" is performed before the addition "3 + 6 = 9," a strategy that is derived from the arithmetic principle of associativity. However, both children and adults make infrequent use of this shortcut and the reasons for this are currently unknown. To uncover these reasons, new sensitive measures of strategy identification and use must first be developed, which was one goal of our research. We built a novel method to detect the time-point when individuals first identify an arithmetic strategy, based on trial-by-trial response time data. Our second goal was to use this measure to investigate the contribution of one particular factor, attention, in the identification of the associativity shortcut. In two studies, we found that manipulating visual attention made no difference to the number of people who identified the shortcut, the trial number on which they first identified it, or their accuracy and response time for solving shortcut problems. We discuss the theoretical and methodological contribution of our findings and argue that the origin of people's difficulty with associativity shortcuts may lie beyond attention.

Benefit of temporal fine structure to speech perception in noise measured with controlled temporal envelopes.

Previous studies have assessed the importance of temporal fine structure (TFS) for speech perception in noise by comparing the performance of normal-hearing listeners in two conditions. In one condition, the stimuli have useful information in both their temporal envelopes and their TFS. In the other condition, stimuli are vocoded and contain useful information only in their temporal envelopes. However, these studies have confounded differences in TFS with differences in the temporal envelope. The present study manipulated the analytic signal of stimuli to preserve the temporal envelope between conditions with different TFS. The inclusion of informative TFS improved speech-reception thresholds for sentences presented in steady and modulated noise, demonstrating that there are significant benefits of including informative TFS even when the temporal envelope is controlled. It is likely that the results of previous studies largely reflect the benefits of TFS, rather than uncontrolled effects of changes in the temporal envelope.