Why is 9+7 harder than 2+3? Strength and interference as explanations of the problem-size effect

In four experiments, the problem-size effect was investigated, using an alphabet-arithmetic task in which subjects verified such problems as A + 2 = C. Problem size was manipulated by varying the magnitude of the digit addend (e.g., A + 2, A + 3, and A + 4). The frequency and similarity of problems was also manipulated to determine the contribution of strength and interference, respectively. Experiment 1 manipulated frequency at low levels of practice and found that strength could account for the problem-size effect. Experiment 2 manipulated frequency at higher levels of practice, and found that strength alone could not account for the problem-size effect at asymptote. Experiment 3 manipulated frequency and similarity and found a substantial problem-size effect at asymptote, suggesting that both strength and interference contribute to the problem-size effect. Experiment 4 manipulated similarity, keeping frequency constant, and found no problem-size effect at asymptote, suggesting that interference alone is not responsible for the problem-size effect. The results are related to findings with number arithmetic.     

Constraints on strategy construction in a speeded discrimination task

In three experiments, subjects reported the identity of a word (above or below) that appeared above or below a fixation point. On some trials, a cue presented 100, 200, 400, 600, 800, or 1,000 msec before the word indicated the relation between position and identity (i.e., whether the dimensions were compatible, e.g., above/above and below/below, or conflicted, e.g., above/below and below/above). On the other trials, the cue was withheld (Experiment 2) or it bore no information about the relation between dimensions (Experiment 1 and 3). In each experiment, the cue reduced reaction time below the level of no-cue or neutral-cue controls, indicating strategic use of the relation between dimensions. Experiments 1 and 2 manipulated the number of potential cues that could occur in a block. A stronger cuing effect was found when one cue could occur (Experiment 2) than when two cues could occur (Experiment 1). Experiment 3 manipulated practice; it revealed that with practice the cuing effect reached asymptote at shorter delays. The asymptote itself did not change. Experiment 4 showed that cue-delay effects were independent of warning interval (warning interval and cue delay confounded in Experiment 1, 2, and 3). The experiments demonstrate construction and utilization of strategies; they show that construction is sensitive to constraints imposed by the subject's goals and abilities and by the structure of the task environment.     

When it hurts to be misled: A Stroop-like effect in a simple addition production task

In four experiments, subjects saw simple addition equations (e.g., 3 + 4 = 9) and produced the sums while ignoring the presented answer. If the presented answer was false, subjects took longer to produce the sum, as compared with when the presented answer was true (Experiment 1), when there was no answer presented (blanks; Experiment 2), when a letter was presented (Experiment 3), and when a symbol was presented (Experiment 4). The results suggest that subjects were unable to ignore the presented answers, which raises problems for theories of arithmetic verification (i.e., deciding whether 3 + 4 = 9 is true or false) that claim that subjects verify equations by first producing the sum and then comparing the produced sum with the presented answer. Our results are more compatible with theories that claim that in verification and production, an arithmetic knowledge base is used in different ways.     

What’s up in mental rotation?

We report two experiments on the influence of head tilt on mental rotation. In Experiment I, subjects decided whether dot patterns were or were not repeated about a line. Their reaction times (RTs) were consistent with the interpretation that they mentally rotated the patterns so that the line was subjectively vertical before making their decisions. When the subjects tilted their heads, the RT functions shifted in the direction of the tilt, indicating that the subjective vertical lay closer to the retinal than to the gravitational vertical. In Experiment II, subjects decided whether singly presented alphanumeric characters in various orientations were standard or backward (mirror-reversed). Again, analysis of their RTs suggested mental rotation to the standard upright, but the function was unaffected by head tilt; in this case, the subjects operated in subjective gravitational rather than retinal coordinates. The choice of retinal or gravitational coordinates may depend on whether the stimuli are interpreted egocentrically or as part of the external world.     

Response set effects in recognition memory

In three experiments, Ss responded to individual digits or letters according to whether or not each was in some prememorized list. There were either two possible responses (yes-no condition) or a single response (yes-only and no-only conditions). With memory sets of one, two, or four digits, RT was a linear function of memory set size. The slope of the function was least under the yes-only condition and greatest under the yes-no condition. Nonspecific practice had little effect on any of the slopes. With memory sets of 4, 8, or 12 letters, the slopes under the yes-only and yes-no conditions did not seem to differ, and practice with specific sets flattened the function considerably in both cases. Overall, the errors under the yes-no condition were mostly false alarms, those under the no-only condition mostly misses, and those under the yes-no condition were divided about equally. The results are interpreted partially in terms of a multiple-observations model of decision time.     

When it helps to be misled: Facilitative effects of increasing the frequency of conflicting stimuli in a Stroop-like task

Three experiments are reported that involve responding to the meaning or position of a word (Above or Below) presented above or below a fixation point. Position and word meaning conflicted (Above/below or Below/above) or were compatible (Above/above or Below/below), and the relative frequency of conflicting trials was varied. Experiment 1 required responses to the word and its position. Compatibility and frequency had no effect in the spatial task, but interacted strongly in the word task: Compatible stimuli were processed faster when conflicting trials were rare (20% conflicting), but conflicting stimuli were processed faster when they were frequent (80% conflicting). Experiments 2 and 3 used the word task only and extended these findings to intermediate (20%, 40%, 60%, and 80% conflicting) and more extreme (10%, 20%, 80%, and 90% conflicting) frequencies, respectively. The advantage for conflicting stimuli when they were frequent was taken as evidence for a strategy involving dividing attention between reported and unreported dimensions.     

On the autonomy of mental processes: a case study of arithmetic

We define a process as autonomous if it can begin without intention, and if it can run on to completion without intention. We develop empirical criteria for determining whether a process can begin without intention, for determining whether it begins in the same way without intention as it does with intention, and for determining whether it can run on to completion without intention once it begins. We apply these criteria to assess the autonomy of the processes underlying simple mental arithmetic--the addition and multiplication of single digits--and find evidence that simple arithmetic may be only partially autonomous: It can begin without intention, but does not begin in the same way without intention as with intention and does not run on to completion without intention. This conclusion suggests there may be a continuum of autonomy, ranging from completely autonomous to completely nonautonomous.     

Costs and benefits of strategy construction in a speeded discrimination task

The costs and benefits of constructing and utilizing a strategy for performing a speeded discrimination task were assessed in terms of time and mental resources. The strategy involved using information about the position a word occupied in order to respond to its identity more rapidly. The resources used to construct and utilize the strategy were assessed from responses to probes presented at various times during the task. In Experiment 1, the probe required a manual response, whereas in Experiment 2, the probe required a vocal response. Strategy construction interfered with the probe task in Experiment 1 but not in Experiment 2, suggesting that strategy construction required resources specific to the manual response system. There was also some evidence that strategy construction required general resources, but the evidence for specific resources was much stronger. The findings are consistent with our previous ideas about how the strategy is represented (Logan, 1980a; Logan & Zbrodoff, 1979). The different results with different responses to the probes point out important limitations on previous studies of probe interference and of cost-benefit analysis, which each focused on a single dimension of cost and benefit. The differences suggest that a multiple-resources approach to probe interference and cost-benefit analysis might be more appropriate.