Results support the effectiveness and durability of the cognitive training interventions in improving targeted cognitive abilities. Training effects were of a magnitude equivalent to the amount of decline expected in elderly persons without dementia over 7- to 14-year intervals. Because of minimal functional decline across all groups, longer follow-up is likely required to observe training effects on everyday function.

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Drawing on various approaches to the study of mathematics learning, Gersten, Jordan, and Flojo (in this issue) explore the implications of this research for identifying children at risk for developing mathematical disabilities. One of the key topics Gersten et al. consider in their review is that of "number sense." I expand on their preliminary effort by examining in detail the diverse set of components purported to be encompassed by this construct. My analysis reveals some major differences between the ways in which number sense is defined in the mathematical cognition literature and its definition in the literature in mathematics education. I also present recent empirical evidence and theoretical perspectives bearing on the importance of measuring the speed of making magnitude comparisons. Finally, I discuss how differing conceptions of number sense inform the issue of whether and to what extent it may be teachable.

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The goals of the present experiment were (1) to determine the function relating rate of presentation of neutral background events to accuracy of signal detection in both a simultaneous (comparative judgment) and a successive (absolute judgment) vigilance task, and (2) to test differential predictions from Posner's pathway inhibition theory and Jerison's elicited observing-rate theory about the effects of heterogeneity in the shape of the neutral events. Data for simultaneous and successive tasks at four event rates (6, 12, 24, and 48 events/min) under both homogeneous and heterogeneous neutral-event conditions showed that the event-rate function (a decline in detections with increments in event rate) was steeper for the successive than for the simultaneous task. Consistent with Jerison's view but not with Posner's, stimulus heterogeneity did not moderate the effects of either time on watch or event rate.