Journal of Experimental Psychology: General 2002, Vol. 131, No. 2, 206 –219

Copyright 2002 by the American Psychological Association, Inc. 0096-3445/02/$5.00 DOI: 10.1037//0096-3445.131.2.206

Reaching While Calculating: Scheduling of Cognitive and Perceptual-Motor Processes Jacqueline C. Shin and David A. Rosenbaum Pennsylvania State University

To address the neglected question of how cognitive and perceptual-motor processes are coordinated, the authors asked participants to move a cursor from one target to another to reveal operators and operands for a running arithmetic task. In Experiment 1 performance on this task was compared with performance on tasks requiring only aiming or only arithmetic. Aiming was faster in the aiming-only task than in the combined task. More importantly, times for steps requiring calculation were equivalent in the combined and arithmetic-only tasks. The results from this and a second experiment suggest that participants slowed their aiming to allow calculations to be completed before subsequent targets were entered. As a whole, the results suggest that cognitive and perceptual-motor processes are coordinated through scheduling.

activity, walking, is carried out to achieve a cognitive goal, completing the thesis. The student could adopt different strategies for coordinating her walking and thinking. She could think about the thesis only during meetings with the professors, or she could think both during the meetings and also while walking between the professors’ offices. If the time to think about the topic for each meeting took somewhat longer than the time to walk to the meeting, it would be advantageous for the student to walk a bit more slowly than usual to arrive at the meeting with completed thoughts. She might be a little late for the meetings but her professors would approve of her thoughtfulness. This example provides a model for a possible research protocol on cognitive and perceptual-motor coordination. In the task we used, instead of asking participants to walk back and forth between offices while completing doctoral dissertations, we asked them to move a cursor back and forth between targets on a computer monitor while performing a running arithmetic task. The critical difference between this task and ones that have been studied in other dual-task situations is that the aiming was performed for the sake of arithmetic. The aiming goals were functionally subordinate to, or “nested in,” the arithmetic goal. Consequently, we called this the nested aiming–arithmetic task. We made a number of predictions about this task on the basis of alternative accounts of how it might be performed. To set the stage for the predictions here, we first preview the task events, then present the alternative models, report two experiments designed to test the models, and end with a discussion of the general implications of our findings.

Although a great deal of research has been devoted to understanding cognitive tasks alone (e.g., mental arithmetic, reading, or problem solving) and perceptual-motor tasks alone (e.g., aiming, reaching, or tracking), few studies have asked how such tasks are coordinated. In everyday life we rarely engage in cognitive processes without also engaging in perceptual-motor activity, and vice versa. We carry out writing movements to express ideas, we hit calculator keys to perform computations, and so on. When perceptual-motor and cognitive activities are performed together, they are usually coordinated to achieve a common goal. The question posed here is how this coordination is achieved. Some of the issues that arise in connection with this question may be illustrated in the following scenario. A student preparing to write a thesis on an interdisciplinary topic such as mathematical psychology has two advisors— one in the Mathematics Department and one in the Psychology Department. The student meets individually with the professors and walks back and forth between their offices while preparing her thesis. Her perceptual-motor

Jacqueline C. Shin and David A. Rosenbaum, Department of Psychology, Pennsylvania State University. This work is based on a doctoral dissertation by Jacqueline C. Shin done at Pennsylvania State University under the supervision of David A. Rosenbaum. The research was supported by Air Force Office of Scientific Research Summer Research Extension Program Grant F49620-93-C-0063 awarded to Jacqueline C. Shin and by National Science Foundation Grant SBR-9308671, a National Institute of Mental Health Research Scientist Development Award, and a Pennsylvania State University Liberal Arts grant awarded to David A. Rosenbaum. Experiment 2 was conducted at Lackland Air Force Base, Texas. We are grateful to Scott Chaiken of Lackland Air Force Base for his support. We also thank Pamela Daly and Shawna Skinner for help with data collection, and Mark Ashcraft, Akira Miyake, and an anonymous reviewer for helpful comments. Correspondence concerning this article should be addressed to Jacqueline C. Shin, who is now at the Department of Psychology, University of Virginia, Gilman Hall, P.O. Box 400400, Charlottesville, Virginia 229044400. E-mail: [email protected] or [email protected]

Task Preview Figure 1 provides a schematic of the experimental setup. The participant used a computer mouse to move a cursor from target to target on a screen. When the cursor entered the target, a number appeared beside it and remained on the screen until the cursor left the target. The participant then moved the cursor into the target on the other side of the screen, where an operator and operand appeared (e.g., ⫺2). The participant continued moving the cursor 206

COGNITIVE AND PERCEPTUAL-MOTOR SCHEDULING

207

back and forth between the left and right targets, noting the operator– operand pair beside each one. At the penultimate step, an equal sign (“⫽”) appeared, and at the last step, a number appeared. The participant ended the trial under the instruction to move the cursor into the True target at the top of the screen if the last number matched the correct running sum or to move the cursor to the False target at the bottom of the screen if it did not. The final target was supposed to be reached in as little time as possible after the trial began.

Models Interference

Figure 1. Sequence of events in a typical experimental trial in the nested task. Circles represent targets. Boxes along the horizontal meridian are sites for sign–numeral pairs. When the trial began, the participant placed the cursor in the center target (as shown by the filled center target). An asterisk appeared in one target circle (here the left one). On bringing the cursor to the left target in Step 1, the participant saw a numeral in the left box. Then, on bringing the cursor to the right target in Step 2, the participant saw an operator– operand pair in the right box. Back-and-forth horizontal movements continued until Step 6, when the participant saw an equal sign, followed in Step 7 by a number that equaled or did not equal the running sum. For this problem, the complete series of sign–numeral pairs was 6 ⫺ 2 ⫹ 4 ⫺ 5 ⫹ 1 ⫽ 4, in which case 4, the number displayed after the “⫽” sign, was the correct sum, which the participant indicated by bringing the cursor to the target adjacent to the True box. The correct sum was shown on half the trials.

How might aiming and arithmetic be coordinated in this nested aiming–arithmetic task? A possible source of answers is interference theory. Models of interference predict dual-task decrements for aiming and arithmetic. The extent of the decrements should depend on how and whether the two tasks share common resources (Kahneman, 1973; Wickens, 1980), common structures (Heuer & Wing, 1984; McLeod, 1980), or bottlenecks (Broadbent, 1957; Keele, 1973; Pashler, 1984; Welford, 1952). Aiming and arithmetic might be slowed in the nested task if they competed for visuospatial processing resources (e.g., if visual-spatial information is used for guiding movement on the one hand and for visually encoding operators and operands on the other). Similarly, aiming and arithmetic might be slowed in the nested task if they competed for access to some information-processing bottleneck (e.g., for initiating movements and for calculating). Although aiming and arithmetic might interfere with each other, it is unclear what specific predictions can be made with respect to how the interference might be played out in the timing and accuracy of responses. The ambiguity stems from the fact that research on dual-task interference has mainly focused on situations in which the component tasks are not purposefully related to each other. This has been true whether the subtasks require discrete responses, as in the psychological refractory period (PRP) paradigm (Kantowitz, 1974; Welford, 1952) or involve continuous perceptual-motor and cognitive responses, as in tracking and mental arithmetic (Wickens & Liu, 1988). The lack of goal relatedness in dual-task situations reflects the desire to use secondary-task methodology to measure resource demands in a wide range of arbitrarily related task ensembles. Teleological subordination of one task to another does not lend itself easily to arbitrary task matching. For discussions of secondary-task methods, see Kantowitz (1974), Kerr (1973), and Ogden, Levine, and Eisner (1979). Another reason why past studies of interference may be less instructive than one might like is that by emphasizing the role of resources or bottlenecks, they minimize the role of strategic factors in dual-task performance, notwithstanding a few exceptions that emphasize skill-based resource allocation (Allport, 1980; Hirst & Kalmar, 1987). Previous research has shown that performers are adept at exercising control over the timing of their mental processes. For example, participants can control the order in which cognitive subtasks (memory search and mental arithmetic) are performed in response to sequencing instructions (Ehrenstein, Schweickert, Choi, & Proctor, 1997), and interference effects in the PRP paradigm vary as a function of instructions about response ordering and stimulus predictability (Greenwald & Shulman,

SHIN AND ROSENBAUM

208

1973). Such findings indicate that a complete theory of performance in tasks with goals and subgoals cannot rely exclusively on interference; it must also include strategic control.

Scheduling A different body of work that may help generate predictions about the nested aiming–arithmetic task concerns scheduling theory. This is a formalized study of algorithms used to sequence operations for optimizing performance goals (Graham, 1978; Schweickert, 1980). A typical application of scheduling theory is arranging manufacturing procedures to maximize output while minimizing cost. In applying scheduling theory to the nested task, one can begin with the assumption that performers time their component actions to maximize accuracy and minimize total time. Adopting a scheduling perspective does not mean rejecting interference. According to scheduling theory, interference can be regarded as a cost to be minimized. Assuming that the amount of interference is influenced by the overlap of structurally incompatible component activities, interference can be reduced by minimizing such overlap as much as possible. The latter idea has precedent in constructs such as the central executive (Baddeley, 1986), resource allocation (Kahneman, 1973), and the supervisory attentional system (Norman & Shallice, 1980). Nonetheless, as Meyer and Kieras (1997a, 1997b) pointed out, despite the importance of these formulations, they do not easily enable detailed quantitative predictions for novel task domains. Meyer and Kieras’s approach emphasized goal-related temporal control, respecting the centrality of interference effects. On the basis of this approach, Meyer and Kieras successfully modeled various PRP phenomena, focusing almost exclusively on the scheduling of “if–then” action contingencies (“productions”). A major assumption in Meyer and Kieras’s work was that performers time their cognitive processes to meet explicit task requirements related to response ordering. The timing is designed to avoid putative bottlenecks such as those associated with response initiation (Kantowitz, 1974; Keele, 1973). In the work reported here we extend this approach to the task of moving for the sake of solving math problems. Arguably, this task is both more complex and more natural than the PRP tasks studied by Meyer and Kieras (1997a, 1997b). Another difference between our approach and the one used by Meyer and Kieras, or indeed used in any traditional application of scheduling theory (see Anderson, Sweeney, & Williams, 1982), is that we allow process times to be lengthened to maximize efficiency of parallel processing. Traditional applications of scheduling theory do not permit process times to be modified. Instead, they only allow variable ordering (with possible overlap) of processes whose mean durations are fixed. Because, as will be seen below, our analysis points to the value of lengthening component times, our research may suggest useful extensions of scheduling theory in domains outside of experimental psychology (e.g., arranging manufacturing procedures). Let us now pursue a scheduling analysis to predict an optimal scheduling strategy for the nested aiming–arithmetic task. The analysis is developed with respect to the times when calculations occur and the degree to which movements and mental arithmetic overlap.

One strategy would be to postpone all calculations until all operators and operands have been displayed. Such a strategy might work with a small number of operator– operand pairs or with a highly predictable sequence of operator– operand pairs. However, it would probably not work when there are many operator– operand pairs whose identities are unpredictable (the situation studied here) because such sequences place heavy demands on working memory (Healy & Nairne, 1985). Another strategy would be to complete calculations one at a time at each step. This method would alleviate the burdens of maintaining and coordinating information from multiple steps in working memory (Charness & Campbell, 1988; Elio, 1986; Wenger & Carlson, 1995; Yee, Hunt, & Pellegrino, 1991). One way this strategy could be implemented would be to calculate and aim in a strictly serial fashion (Figure 2, top rows of Panels A, B, and C). The participant could move the cursor into a target, encode the associated operator and operand, update the running arithmetic sum, and then move to the next target. With this strictly serial approach, the time for each step (i.e., the time to enter one target after entering the previous target) would approximate the sum of the times for encoding, calculating, and aiming.1 The strategy just presented would be inefficient because it would result in dead time for aiming during arithmetic and dead time for arithmetic during aiming. Insofar as it may be possible to aim and do arithmetic simultaneously, it would be more efficient to let the two processes run in parallel. If the time to calculate happened to equal the time to aim (Figure 2A), parallel calculation and aiming (bottom row of Figure 2A) would allow step time (i.e., the time to enter one target after entering the previous target) to remain within the sum of the times for encoding and calculating (or equivalently, within the sum of the times for encoding and aiming). Alternatively, if the time to calculate happened to be less than the time to aim (Figure 2B), then, as shown in the bottom row of Figure 2B, parallel calculation and aiming would allow step time to equal the sum of encoding time plus aiming time; step time in this case would be smaller than the sum of encoding time, calculating time, and aiming time. Finally, if the time to calculate happened to exceed the time to aim (Figure 2C), parallel calculation and aiming would allow step time to equal the sum of the times for encoding and calculating provided aiming time was stretched by at least the difference between calculation time and normal aiming time (middle row of Figure 2C). The latter strategy would be analogous to the one used by the deliberately dawdling doctoral candidate.2 A final possibility is the one shown in the bottom row of Figure 2C. Here, in contrast to what appeared in the middle row of Figure 2C, aiming and calculating do not begin together. Instead, aiming begins after calculation starts. With this strategy, the observed times— dwell time and movement (or aiming) time—are both

1 Throughout this article we will use the terms aiming time and movement time interchangeably. We will also use the terms arithmetic time and calculation time interchangeably. 2 In this, as in all the models considered here, it is assumed that calculation never begins before encoding ends.

COGNITIVE AND PERCEPTUAL-MOTOR SCHEDULING

Figure 2. Scheduling models for the nested aiming–arithmetic task and corresponding predictions about dwell time (DT) and aiming, or movement, time (MT). A: Calculating and aiming take approximately equal times and are carried out either serially (top row) or in parallel (bottom row). B: Calculating takes less time than aiming, and calculating and aiming either are carried out serially (top row) or are done in parallel (bottom row). C: Calculating takes more time than aiming, and calculating and aiming either are carried out serially (top row) or are done in parallel such that aiming and calculating begin and end together (middle row), or aiming begins after calculating starts and aiming and calculating end together (bottom row). In Panel C, aiming time (MT) is stretched in the two parallel methods (middle and bottom rows) relative to its baseline duration. Note that in all these scenarios it is assumed that calculation begins after encoding is completed. Encoding time is the same in all cases. Step time is approximately equal to DT ⫹ MT. See text for details.

stretched relative to their baseline levels when no calculation is required.3

Experiment 1 In the first experiment, we sought to distinguish among these scheduling strategies. On the basis of the foregoing analysis, we hypothesized that participants would adopt a parallel strategy rather than a serial strategy. We also hypothesized that if calculation time was longer than aiming time and that if there was a

209

preference to keep dwell time and movement time relatively close to their baseline values when parallel processing was used, the evidence would favor the bottom row of Figure 2C rather than the middle row of Figure 2C. To test these competing predictions, we asked participants to perform the nested aiming–arithmetic task as well as an arithmeticonly task and an aiming-only task. In the arithmetic-only task, we asked participants to perform mental arithmetic, but instead of making controlled movements of the cursor back and forth between the targets, they merely had to nudge the mouse to the left or right or, in the final step, up or down. Participants performing this task saw the same kinds of stimuli as in the nested task and had the same calculation requirements. We assumed that the times to move from target to target in the arithmetic-only task would provide an estimate of the time needed to encode the operands and operators as well as the time to calculate, as required to distinguish among the models in Figure 2. Note that this amounts to a pure-insertion assumption, which, to be sure, is a strong assumption (Sternberg, 1969). Because our subsequent inferences rely heavily on this assumption, they must be drawn cautiously. In the aiming-only task, participants moved the cursor back and forth between the targets, as in the nested task, and the displays were identical to those shown in the nested task. While performing the aiming-only task, participants had to do simple number monitoring. At each step, they had to keep track of the last numeral displayed with a plus (⫹) sign. If the participants saw ⫹5, ⫺6, ⫺4, ⫹7, for example, they were supposed to keep the numeral 5 in mind until the 7 appeared. If the numeral shown after the equal sign matched the last numeral shown with a plus sign, the participants were supposed to move the cursor into the True target; otherwise they were supposed to move the cursor into the False target. Pilot work showed this to be an easy task. The reason for including the number monitoring requirement was to make sure participants attended visually to the operands and operators as they had to in the arithmetic-only and aiming–arithmetic tasks. We assumed that the times to move from target to target in the arithmetic-only task provided an estimate of the time needed to encode the operands and operators plus the time to move, as needed to choose among the models in Figure 2. Once again, this relies on the pure-insertion assumption.

Method Apparatus and Stimuli Participants used a computer mouse to move the cursor (a crosshair) from one target to the next. The position of the cursor was mapped directly to the position of the mouse. The left and right targets to which participants aimed were circles whose centers were 3.3 cm apart on the computer monitor, or 5.0° apart in visual angle when viewed by the participant, whose eyes were approximately 75 cm from the screen. The targets were 3 mm wide or 0.4° visual angle in radius. Given the distance between the centers of the left and right targets, it was possible to use Fitts’ Law (Fitts, 1954) to calculate the index of

3

When we say aiming time is stretched we do not mean to imply that the kinematics of the aiming movement perfectly scales with duration. Such scaling is not usually observed (Gentner, 1987).

210

SHIN AND ROSENBAUM

difficulty for the movements and to predict the time needed to move between them when movement time was minimized.4 As seen in Figure 1, squares were displayed to the left of the left target and to the right of the right target. The squares housed the operators and operands. The squares were 0.9 cm ⫻ 0.9 cm (0.7° ⫻ 0.7° in visual angle). Inside each square appeared the information needed for the arithmetic calculations when the cursor went into the corresponding target for the first time in each step (see below). The centers of the squares were 1.1 cm (0.8°) from the centers of their respective targets. There were also top and bottom targets, whose centers were 2.5 cm (1.9°) apart. Above the top square and below the bottom square were rectangles, 1.4 cm ⫻ 0.9 cm (1.1° ⫻ 0.7°), in which appeared the words True and False, respectively, throughout the entire trial. The centers of the rectangles were 1.1 cm (1.7°) from the centers of the top and the bottom targets. Each time the cursor entered a target after leaving the center of the screen on the first step or after leaving another target in subsequent steps, arithmetic information was displayed in the adjacent square. When the cursor was moved out of a target, the arithmetic information disappeared and did not reappear if the cursor reentered the target before going to the other target. Given this arrangement, it was important for participants to dwell in each target long enough to encode the information beside it. The position of the mouse, as reflected in the spatial location of the cursor on the computer screen, was recorded with a Macintosh IIsi computer used to run the entire experiment. The spatial resolution of the computer monitor was 1 pixel ⫽ 0.35 mm ⫻ 0.35 mm, or 0.05° ⫻ 0.05°. The temporal resolution was 60 Hz. The mouse only had to be slid in this experiment; no clicking of the mouse was required.

Procedure Participants performed the nested task, the aiming-only task, and the arithmetic-only task during each of two 1-hr sessions on consecutive days. In the nested task the sequence of events was as follows (see Figure 1): At the start of the trial a display appeared on the screen consisting of the center target, the left and right targets with empty rectangles beside them, the top and bottom targets with adjacent rectangles containing the words True and False, and the cursor. The trial began when the participant moved the cursor into the center target, at which time an asterisk (*) appeared in the left or right target to indicate the first target to which the participant was supposed to move. Left and right targets were used randomly as the first target except for the constraint that they be used equally often. When the participant moved the cursor out of the center target, that target disappeared for the remainder of the trial. When the participant moved the cursor into the first target, the first numeral in the sequence appeared in the adjacent square. When the participant moved the cursor into the opposite left or right target, the next stimulus (an operator and operand) appeared in the square beside it. The participant continued moving the cursor back and forth between the left and right targets. In the second through fifth steps, a plus (⫹) or minus (⫺) sign appeared with a numeral. On reaching the sixth target, an equal sign (⫽) appeared, and on reaching the seventh target, a possible solution was displayed, at which time the participant was supposed to move the cursor to the True target if the solution was correct or to the False target if it was incorrect. Reaching the True or False target terminated the trial. The displayed numbers were all integers. The number presented at Step 1 was between 6 and 11, and the numbers presented at Steps 2 through 5 were between 2 and 7. The intermediate and final running totals were between 2 and 15. The same number was never presented on two consecutive steps. The number presented at the last step equaled the actual solution of the arithmetic sequence on a random half of the trials in each block. When the last number differed from the actual solution, it was greater or less than the actual solution equally often in a block, and differed randomly by ⫾1, ⫾2, or ⫾3 from the actual solution. Two of the operators were addition and the other two were subtraction. The operators were presented in random order.

After each trial, participants received feedback about the accuracy of their True and False responses. At the end of each block of 16 trials they were given feedback both about their true–false accuracy and their trial completion times for correct trials in the just-completed block. The endof-block feedback was provided to reinforce instructions emphasizing both speed and accuracy. At the end of each block of 16 trials, participants also received their best block score up to that point in the current set of two blocks (i.e., within the condition being tested). Participants were encouraged to better their scores within each condition. Participants were never told that dwell times and movement times were recorded separately. The only time referred to was the total time for the trial. In the aiming-only task participants moved the cursor back and forth between the targets as in the nested task, and the display of information was identical to what was used in the nested task. As mentioned earlier, participants in the aiming-only task were asked to do a number-monitoring task. They were asked to bring the cursor into the True target if the number that was displayed after the equal sign matched the last number displayed with a plus sign. Conversely, they were asked to bring the cursor into the False target if the number that was displayed after the equal sign did not match the last number displayed with a plus sign. In the arithmetic-only task participants were supposed to perform mental arithmetic as in the nested task, but instead of making controlled movements of the cursor back and forth between the targets, they merely had to nudge the mouse (at least 5 pixels). As soon as the mouse was nudged in the appropriate horizontal direction, in all but the final step the next target turned black and the arithmetic information for that step appeared in the square beside it. The arithmetic information remained on the screen until the mouse was nudged in the opposite direction, at which time the arithmetic information for the next step appeared immediately in the adjacent square and the information from the previous step disappeared. In the final step, a slight nudge of the mouse upward or downward caused the cursor to jump up to the True box or down to the False box, respectively. Pilot work showed that nudging the mouse in the cardinal directions (left, right, up, or down) was very easy. During the arithmetic-only (nudging) task, arithmetic information was always present on the computer monitor. This contrasted with what happened in the nested and aiming-only tasks, in which arithmetic information was displayed only when the cursor occupied a target for the first time. This difference was unavoidable given the mechanics of the nudging task. In all other respects, the arithmetic-only, aiming-only, and nested tasks were the same.

Design and Data Analysis Session 1 was for practice only. In this session, participants practiced each of the tasks in two blocks of 16 trials each, and then the same two blocks of 16 trials each were practiced again. The order of tasks was counterbalanced across participants. Participants were coached throughout Session 1 to encourage them to maximize speed and accuracy. Because of the ensuing interruptions, the data from the first session were not used in

4 Fitts’ Law (Fitts, 1954) says that the time to move from one target to another increases with the distance between the targets and decreases with the targets’ widths, or more specifically, MT ⫽ k1 ⫹ k2log2(2D/W), where MT is movement time, D is the distance to be traveled, W is the width of the target, and k1 and k2 are positive constants (see Meyer, Smith, Kornblum, Abrams, & Wright, 1990, for review). Because log2(2D/W) specifies the perceptual-motor demands of an aiming movement, it is often referred to as the index of difficulty. The index of difficulty for the aiming movements required in Experiment 1 was 3.58. It is worth noting that Fitts’ Law has been supported in previous studies that used the aiming task studied here—moving a computer mouse to bring a cursor from one screen target to another (MacKenzie, 1992).

COGNITIVE AND PERCEPTUAL-MOTOR SCHEDULING later analyses. In Session 2 all participants performed the tasks in the reverse order of what they did in Session 1. Thus, each participant performed a total of 64 trials per task in each experimental session. The data from Session 2 comprised the only data analyzed for this presentation. Measures were taken both of overall trial performance and of step-bystep performance. The measures of overall trial performance were the proportion of True or False responses and the trial’s total time, defined as the time from when the cursor entered the center target to when it entered a True or False target. The measures of step-by-step performance were step time, dwell time, and movement time. Step time for the ith step was defined as the time between when the cursor first entered target i and when it first entered target i ⫹1 (i ⫽ 1, 2, . . ., 7). Dwell time for the ith step was defined as the time between the cursor’s first arrival and first departure from target i (i ⫽ 1, 2, . . ., 7). Movement time for the ith step was defined as the time the cursor traveled from target i ⫺ 1 to target i (i ⫽ 1, 2, . . ., 8). Note that the movement time for Step 1 included the time the cursor dwelled in the center target (Target 0) plus the time for the cursor to move into the first left or right target (Target 1). The movement time for Step 8 was the time to move to the True or False target. In the nested and aiming-only tasks, step time did not necessarily equal the sum of dwell time and movement time because dwell time excluded the time during which the cursor left and reentered a target without first entering the other target. These extra “detour” times were usually very short. A final point about the dependent measures is that dwell time and movement time were defined only for the nested and aiming tasks, not for the arithmetic task. For the arithmetic task, only step time was defined. For all three tasks, we were primarily interested in step times for those targets in which an operator– operand pair (e.g., ⫹3) appeared. We referred to these as calculation steps. For step times and dwell times, the calculation steps were Steps 2 through 5. For movement times, the calculation steps were Steps 3 through 6. We evaluated the effects of task on the measures outlined above using analyses of variance (ANOVAs) and paired-comparisons t tests whose critical values for significance ( p ⬍ .05 by default) were adjusted according to the Bonferroni procedure when multiple t tests were needed. For time measures, we used participants’ median times.

Participants Nine undergraduate and three graduate students, all from Pennsylvania State University and all naı¨ve to the aims of the research, participated for pay.

Results Consistent with our treatment of Session 1 as a coaching session only, we now report the results of the second session. For the dwell times and movement times of the nested and aiming tasks, the data analysis excluded trials in which participants made left or right moves rather than moving to the True or False target after the possible solution was shown. For the nested and aiming tasks, such perseverative movements occurred in 17 trials (1.1% of all trials). The method of data collection did not allow for identification of perseverative movements in the arithmetic-only task. Analyses of time measures only included trials in which correct True or False responses were given.

Overall Trial Performance The results reviewed in the present section pertain to the two measures of overall trial performance: proportion correct true– false judgments and total time. A single-factor repeated measures ANOVA showed that proportion correct varied among tasks, F(2,

211

22) ⫽ 11.88, MSE ⫽ .001, p ⬍ .001. T tests revealed that proportion correct was higher in the aiming-only task (M ⫽ .99, SE ⫽ .003) than in the arithmetic task (M ⫽ .95, SE ⫽ .012), t(11) ⫽ 3.80, p ⬍ .01. Likewise, proportion correct was higher in the aiming task than in the nested task (M ⫽ .94, SE ⫽ .009), t(11) ⫽ 5.29, p ⬍ .001. Proportion correct did not differ significantly between the arithmetic and nested tasks, t(11) ⫽ .76, p ⬎ .40. Regarding total time, an ANOVA revealed a main effect of task, F(2, 22) ⫽ 22.25, MSE ⫽ 266,278, p ⬍ .0001, such that total time was shortest in the aiming task (M ⫽ 5,026 s, SE ⫽ 262 s), longest in the nested task (M ⫽ 6,427 s, SE ⫽ 393 s), and intermediate in the arithmetic task (M ⫽ 5,630 s, SE ⫽ 388 s). Paired comparisons revealed that total time was longer in the nested task than in the arithmetic and the aiming tasks (ts ⬎ 6.00, ps ⬍ .001). The difference between the aiming-only and arithmetic-only tasks fell short of the Bonferroni criterion, t(11) ⫽ 2.33, p ⬍ .05.

Step-by-Step Performance Step time, dwell time, and movement time, plotted as a function of step number, are shown in Figure 3. Analyses focused on the calculation steps for the three measures: Steps 2 through 5 for step times and dwell times, and Steps 3 through 6 for movement time. Regarding step time, a Task (three tasks) ⫻ Step Number (Steps 2 through 5) ANOVA showed that the interaction between task and step number was significant, F(6, 66) ⫽ 5.72, MSE ⫽ 2,272, p ⬍ .001. Whereas step time hardly depended on step number in the aiming-only task, it depended on step number in the arithmetic and nested tasks. T tests adjusted according to the Bonferroni procedure showed that for Steps 2 through 5, step times in the aiming-only task were significantly shorter than in the other two tasks but that step time in the nested and arithmetic tasks were not significantly different from one another. Regarding dwell time, a Task (nested, aiming) ⫻ Step Number (Steps 2 through 5) ANOVA revealed a significant interaction, F(3, 33) ⫽ 3.25, MSE ⫽ 881, p ⬍ .05, such that dwell times were longer in the nested task than in the aiming-only task. As with step time, dwell time varied with step number in a more pronounced way in the nested condition than it did in the aiming condition. Finally, regarding movement time, a Task (nested, aiming) ⫻ Step Number (Steps 3 through 6) ANOVA revealed a significant interaction between task and step number, F(3, 33) ⫽ 6.30, MSE ⫽ 444, p ⬍ .01, such that movement times were longer in the nested task than in the aiming-only task and, as for step time and dwell time, there was a more marked effect of step number in the nested task than in the aiming task.

Discussion The results of Experiment 1 allowed us to discriminate among the models introduced earlier. The results were inconsistent with the first model we considered, which said that all calculations were postponed until all operands and operators had been displayed. Contrary to this model, step times were not longer in the final one or two steps of the nested and arithmetic-only conditions than in the earlier steps. Furthermore, step times were longer in the nested and arithmetic-only tasks than in the aiming task.

212

SHIN AND ROSENBAUM

moving had occurred serially, the time to do all three would have exceeded the time to do any two. Instead, step time in the nested task approximated the time for the arithmetic-only task, as predicted by the parallel models. Which of the parallel models was best supported? To address this question, it is useful to recall that the parallel models differed with respect to the relative durations of calculation and aiming. According to one parallel model (Figure 2A) calculating and aiming would take the same time, according to another parallel model (Figure 2B) aiming would take more time than calculating, and according to a third parallel model (Figure 2C) calculating would take more time than aiming. As seen in Figure 3A, step times were longer in the arithmetic-only task than in the aimingonly task, which suggests that calculation took longer than aiming, consistent with the model in Figure 2C. Two parallel models appear in Figure 2C, so it is useful to distinguish between them. The middle row of Figure 2C shows a parallel model in which aiming and calculating start and end together, whereas the bottom row of Figure 2C shows a parallel model in which calculating starts before aiming begins but calculation and aiming end simultaneously. The data support the second of these models. This model, shown in the bottom row of Figure 2C, was supported for three reasons. First, step time at the calculation steps was approximately equal in the nested and arithmeticonly conditions (Figure 3A). Second, dwell time (Figure 3B) was longer in the nested condition than in the aiming-only condition. Third, movement time was also longer in the nested condition than in the aiming-only condition (Figure 3C). None of the other models considered here was consistent with these three outcomes.

Experiment 2

Figure 3. Step time (A), dwell time (B), and movement time (C) as a function of step number for each task in Experiment 1. Each value is a mean over participants’ medians. Error bars represent ⫾1 SE.

Given that calculations were apparently done online, how were they coordinated with aiming? The data let us rule out all of the serial models shown in Figure 2 (i.e., the top model in each of the three panels in Figure 2). Contrary to these models, step time in the nested task did not exceed the longer of the time for aiming only, which required encoding and moving, or arithmetic only, which required encoding and calculating. If encoding, calculating, and

As just indicated, the first experiment yielded results consistent with the parallel scheduling strategy, which we conceived as being optimal for minimizing time and errors. The optimality of the strategy remained to be checked, however. Therefore, in Experiment 2 we tested whether the parallel scheduling strategy would in fact result in better performance than a serial scheduling strategy. We did this by forcing participants in some conditions to do all their arithmetic while dwelling at the target before initiating the movement to the next target. This method enforced a serial strategy. Participants in other conditions were allowed to adopt whatever scheduling strategy they wished, as in Experiment 1. On the basis of the results of the first experiment, we expected participants to adopt a parallel scheduling strategy in the ad-lib condition. A related goal of Experiment 2 was to test a different account of the movement slowing observed in Experiment 1. Recall that the model we favored, embodied in the bottom row of Figure 2C, claimed that movement was slowed by concurrent math. To contrast this model with the one introduced next, we refer to the already considered model as the online-slowing hypothesis. The alternative account, which we call the indirect slowing hypothesis, claimed that arithmetic occurred only during dwell time and interfered with movement planning, causing movements to be slower than they would be otherwise. The indirect slowing hypothesis was worth considering because preparatory processes are known to occur prior to aiming movements (e.g., Rosenbaum, Vaughan, Barnes, & Jorgensen, 1992) and movement preparation can be interfered with in dual tasks (Netick & Klapp, 1994; Van Galen,

COGNITIVE AND PERCEPTUAL-MOTOR SCHEDULING

Smyth, Meulenbroek, & Hylkema, 1989). The indirect slowing hypothesis predicted that movement time would be longer in the nested task than in the aiming-only task even when arithmetic had to be completed during dwell time. By contrast, the online-slowing hypothesis predicted that movement time would be longer in the nested task than in the aiming-only task only when arithmetic could extend beyond dwell time.

Method Procedure Each participant was randomly assigned to one of three target exit groups (except for the constraint that there be an equal number of participants in each group): (a) an even-or-odd exit group; (b) a thick-or-thin exit group; or (c) a no-decision group. Participants within each group performed all three of the primary tasks, namely, the nested task, the arithmetic-only task, and the aiming-only task. For participants in the even-or-odd exit group, when they did the nested task they had to perform all their arithmetic during dwell time. Thus, these participants were forced to adopt a serial scheduling strategy with respect to completion of math and movement. Every time participants in this group moved the cursor into the target at Steps 1 through 6, the top or bottom half of the target thickened unpredictably (see Figure 4). The cursor then had to be moved out of the target through the thick or thin half depending on the even– odd status of the sum at that step. If the current result was even, the cursor was supposed to go through the thick half, but if the current result was odd, the cursor was supposed to go through the thin half. When the cursor exited the target, the thickened line returned to its normal thickness. When the same participants did the arithmetic-only task, they had to nudge the joystick up or down to direct the cursor through the thick or thin side of the target on the basis of whether the current sum was even or odd. Finally, when the same participants did the aiming-only task, they had to make the cursor exit the target through the thick side if the last number displayed with a plus sign was even or through the thin side if the last number displayed with a plus sign was odd. For participants in the thick-or-thin target exit group, half of the target was thickened unpredictably as in the even-or-odd target exit group, but participants were always supposed to move the cursor out of the thick half. This served as a control condition for the even-or-odd target exit condition. Finally, in the no-decision exit condition, the thickness of the target remained constant throughout the trial, as in Experiment 1, and participants could move the cursor out of any part of the target, as in the first experiment. This condition, like the thick-or-thin target exit condition, allowed participants to freely adopt a serial or parallel scheduling strategy.

Apparatus and Stimuli Experiment 2 was performed at Lackland Air Force Base (Texas) rather than at Pennsylvania State University. As a result, it used different equip-

213

ment, namely, IBM-compatible personal computers rather than a Macintosh IIsi. As a result, there were some differences in the display and manipulandum. The visual display was larger than in Experiment 1. The radius of the targets was 2.3 cm (2.2° in visual angle when viewed from 60 cm). The distance between the centers of the left and right targets was 16.4 cm or 15.6° in visual angle, and the distance between the centers of the top and bottom (True and False) targets (which were the words True and False themselves without surrounding rectangles) was 17.2 cm or 16.3° in visual angle. The distances between the centers of the horizontal targets and the adjacent rectangles (each 3.4 cm ⫻ 4.2 cm or 3.2° ⫻ 4.0° in visual angle) were 4.0 cm (3.8° in visual angle). The index of aiming difficulty (see Footnote 3) was 2.89. The cursor (radius ⫽ 0.9 cm or 0.9° in visual angle) was controlled with a joystick instead of a computer mouse. As in Experiment 1, the position of the cursor was mapped to the position of the joystick. Symbols (⫹, –, and ⫽) and numerals were presented as in Experiment 1, but at the beginning of each trial, the cursor was automatically centered on the screen when the joystick was centered. There was no center target, and the trial began when the participant pressed a joystick button, although at no subsequent time in a trial did the joystick button have to be pressed, similar to the no-click requirement of Experiment 1. The procedure for displaying arithmetic information in the arithmetic task differed slightly from what it was in Experiment 1, where arithmetic stimuli were always present on the screen at one target or the other depending on the last nudged position of the mouse. In the arithmetic task of Experiment 2 participants caused arithmetic stimuli to appear by letting the joystick come to rest at its spring-loaded center position. Each time participants wanted to bring up the next stimulus they had to nudge the joystick toward the new target and then let it return to its resting, center position. During this brief back and forth gesture, the stimuli next to the previous target disappeared.

Instructions and Feedback As in Experiment 1, participants were instructed to perform all trials as quickly and accurately as possible. Participants in the even-or-odd and thick-or-thin target exit groups were told that a trial was scored correct only if the true–false response was accurate and if the cursor was moved out of the appropriate half of the target in all six steps in which the target thickness was varied. If the cursor was moved out of the target through the central portions of the target boundary (0.5° in visual angle) rather than through the top or bottom half, the step was scored as incorrect. If the participant first moved the cursor out of the incorrect half of the target and subsequently corrected this before moving to the next target, exiting that step was scored as correct. In the no-decision condition, all steps were scored as correct without regard to how the cursor left the target. At the end of each trial, feedback was given about whether the true–false response was correct and about the number of appropriately exited targets. For trials on which the true–false response was correct and all steps were exited correctly, total time was also displayed. At the end of the block, the number of accurate trials in the block and the mean total time for the correct trials were displayed.

Design

Figure 4. Sample display for the even-or-odd target exit group and the thick-or-thin target exit group in Experiment 2. The small circle with a cross represents the cursor. Participants in the even-or-odd exit group had to complete their math before moving, but participants in the thick-or-thin exit group and no-decision group did not.

Each task was presented in two consecutive blocks of 12 trials in each of four sets of blocks. Thus, each participant performed a total of 96 trials per task. The order of tasks was balanced over participants. For a given participant, the tasks were presented in the same order in the first two sets, and the order was reversed in the last two sets. Participants were given a short rest after the second set of blocks. Data were analyzed only for the last two sets of blocks (48 trials per task for each participant) in order to focus on the scheduling strategy established after practice.

SHIN AND ROSENBAUM

214 Participants

Following standard Air Force research procedure and allowing correlation analyses between performance in this experiment and performance in a battery of basic timing, working memory, and induction abilities tests (administered after the tasks described above and not reported here), we included a large number of participants in the study. In groups of 31 each, 182 Air Force basic trainees participated in the experiment in single experimental periods that lasted approximately 3 hr.

Results In this section we report on the measures of overall trial performance and step-by-step performance.5

Overall Trial Performance Proportions of correct true–false responses are presented in Table 1 for each primary task (aiming-only, arithmetic-only, or nested) and each target exit group. An ANOVA showed that the primary Task ⫻ Target Exit Group interaction was not significant ( p ⬎ .40), although the main effect of target exit group was, F(2, 160) ⫽ 7.25, MSE ⫽ 0.0181, p ⬍ .01. T tests revealed that proportion correct was higher in the even-or-odd target exit condition than in the thick-or-thin movement condition, t(102) ⫽ 3.48, p ⬍ .001, and the no-decision condition, t(102) ⫽ 3.89, p ⬍ .001, although the latter two conditions did not differ significantly, t(116) ⫽ .15, p ⬎ .80. The main effect of primary task was also significant, F(2, 320) ⫽ 9.45, MSE ⫽ 0.0052, p ⬍ .001. As revealed by t tests between tasks, proportion correct was higher for the aiming task than for the arithmetic task or the nested task (ts ⬎ 30, ps ⬍ .01), although the latter two tasks did not differ significantly from each other ( p ⬎ .10). Total times for each combination of target exit group and primary task (see Figure 5) were evaluated with an ANOVA that yielded a significant interaction between the two factors, F(4, 320) ⫽ 28.55, MSE ⫽ 2,408,053, p ⬍ .0001. Whereas total time was always shortest in the aiming-only condition, total time was longer in the nested task than in the arithmetic-only task within the even-or-odd and thick-orthin target groups; however, total time was shorter in the nested task than in the arithmetic-only task for the no-decision group. Separate ANOVAs for each primary task (nested, aiming-only, or arithmeticonly) revealed main effects of target exit group for each task (Fs ⬎ 36, ps ⬍ .0001) such that total time was longer in the even-or-odd exiting group than in the other two groups (all pairwise comparisons had ps ⬍ .0001). Total time was longer in the thick-orthin exiting group than in the no-decision group for the aiming-only and nested tasks ( ps ⬍ .0001); the same trend was found for the arithmetic-only task ( p ⬍ .05). Table 1 Mean and Standard Error of Proportion of Correct True–False Responses for Each Group and Task in Experiment 2 Group Even or odd

Thick or thin

No decision

Task

M

SE

M

SE

M

SE

Nested Aiming Arithmetic

.95 .96 .94

.01 .01 .01

.91 .93 .92

.01 .01 .01

.90 .93 .91

.01 .01 .01

Figure 5. Total time for each target exit group (even or odd, thick or thin, or no decision) and task (nested, aiming, or arithmetic) in Experiment 2. Participants in the even-or-odd exit group had to complete their math before moving, but participants in the thick-or-thin exit group and nodecision group did not. Each value is a mean over participants’ medians. Error bars represent ⫾1 SE.

5

Only trials that satisfied the following criteria were included in the analyses of time measures: (a) the True–False response had to be made at the appropriate serial position in the sequence, that is, after the seventh step; (b) each step in the sequence had to be completed within 60 s; and (c) the trial had to be scored as correct (i.e., the cursor was moved out of all six targets from the appropriate side and the True–False response was correct). For 17 of the 182 participants, at least one set of blocks did not have any observations satisfying these criteria for at least one task; the breakdown was 1 in the no-decision group, 1 in the thick-or-thin group, and 15 in the even-or-odd group. We excluded these participants’ data from subsequent analysis. Also excluded from the analyses were data from 2 other participants who did not complete the experiment (1 in the thick-orthin group and 1 in the even-or-odd group). Owing to these exclusions, analyses were conducted on data from the remaining 163 participants, 45 of whom were in the even-or-odd group, 59 of whom were in the thickor-thin group, and 59 of whom were in the no-decision group. For these 163 participants, analyses of the proportion correct True–False responses were conducted for trials which both satisfied criteria (a) and (b) and satisfied the criterion that the participant moved the cursor out of all six targets from the appropriate side. These included 94.9% of trials in the even-or-odd group, 70.8% of trials in the thick-or-thin group, and 86.1% of trials in the no-decision group. An ANOVA testing the effect of target exit condition on the proportion of trials satisfying these criteria revealed that the proportion varied with target exit group, F(2, 160) ⫽ 56.30, MSE ⫽ .0133, p ⬍ .0001, such that the differences among all three groups were statistically significant ( ps ⬍ .001) using the Newman-Keuls procedure to test all pairwise comparisons. Although the proportions of excluded participants and trials were relatively large, this was perhaps inevitable given that we had to restrict our attention to those trials in which we could be sure participants adopted the scheduling or calculation strategy required of them. The fact that the number of participants whose data were excluded was disproportionately large in the even-or-odd group suggests that the strategy required in that group—serial rather than parallel scheduling—was unnatural. Note, however, that the fewest trials were discarded for the even-or-odd group. It is possible that by excluding many participants from this group, we were left with a disproportionately large number of “careful” participants.

COGNITIVE AND PERCEPTUAL-MOTOR SCHEDULING

Step-by-Step Performance Figure 6 shows step times, dwell times, and movement times for individual steps in all conditions of Experiment 2. To keep the

Figure 6. Step time (A), dwell time (B), and movement time (C) as a function of step number for each target exit group (even or odd, thick or thin, or no decision) and primary task (nested, aiming, or arithmetic) in Experiment 2. Participants in the even-or-odd exit group had to complete their math before moving, but participants in the thick-or-thin exit group and no-decision group did not. Each value is a mean over participants’ medians. Error bars represent ⫾1 SE.

215

graphs uncluttered, the only plotted points are for the steps of primary interest—the calculation steps in the nested and arithmetic-only tasks or their analogues in the aiming-only task. In the following, we consider step times first, dwell times second, and movement times third. Step times. In all tasks, step times were defined as the time between when the cursor entered the target at a given step to when the cursor entered the next target. A small complication, attributable to the use of the joystick, was that the tasks differed with respect to the type of movement they employed. Whereas the nested and aiming-only tasks required a unidirectional movement at each step, the arithmetic-only task required a bidirectional back-and-forth movement. That is, in the arithmetic-only task the joystick had to be moved away from the center, effectively erasing the stimulus from the previous target, and then back into the center, effectively starting the display for the following target. The second of these movements, that is, the center-bound movements, were very quick, lasting on average 124 ms. This secondary movement did not show significant interactions between step number and exiting requirement. Therefore, we removed these secondary movements in the arithmetic task from the analysis of step times to make the step times functionally comparable to the others (i.e., making them all unidirectional movements). Several important results are evident from the pattern of step times (Figure 6, Panel A). First, step times were shortest in the no-decision target exit group, intermediate in the thick-or-thin target exit group, and longest in the even-or-odd target exit group. Moreover, in the thick-or-thin and even-or-odd groups, step times were shortest when the primary task was aiming only, intermediate when the primary task was arithmetic only, and longest when the primary task was nested aiming and arithmetic. By contrast, in the no-decision group, step times were shortest when the primary task was aiming only, but there was no difference between step times when the primary task was arithmetic-only or when the primary task was nested aiming and arithmetic. The latter pattern of step times in the no-decision group is similar to the pattern for corresponding step numbers in Experiment 1 (see Figure 3). The foregoing impressions were confirmed statistically. An omnibus ANOVA that evaluated the effects of Target Exit Group ⫻ Primary Task ⫻ Step Number yielded a significant interaction between target exit group and primary task, F(4, 320) ⫽ 19.69, MSE ⫽ 289,122, p ⬍ .0001, reflecting the different separations of means for the nested, arithmetic, and aiming tasks in the even-or-odd, thick-or-thin, and no-decision exit conditions. There was also a significant interaction between primary task and step number, F(6, 960) ⫽ 27.29, MSE ⫽ 23,794, p ⬍ .0001, reflecting the greater dependence of step time on step number in the nested and arithmetic tasks than in the aiming task, as was also observed in Experiment 1. The interaction between target exit group and step number was not significant, F(6, 480) ⫽ 1.49, MSE ⫽ 36,327, p ⬎ .10. ANOVAs confined to each target exit group showed main effects of primary task for each target exit group. Furthermore, paired comparisons among tasks indicated that step times were similar for the arithmetic and nested tasks in the no-decision exiting condition ( p ⬎ .30), as was observed in Experiment 1. By contrast, all three primary tasks differed significantly from each other in the thick-or-thin and the even-or-odd exiting conditions (ts ⬎ 40, ps ⬍ .0001).

SHIN AND ROSENBAUM

216

Dwell times. Regarding dwell time (Figure 6, Panel B), an omnibus ANOVA that evaluated the effects of Target Exit Group ⫻ Primary Task (nested and aiming-only because dwell times per se were undefined for arithmetic-only) ⫻ Step Number showed that the three-way interaction was not significant ( p ⬎ .20). However, the interaction between target exit group and primary task was, F(2, 160) ⫽ 27.09, MSE ⫽ 259,380, p ⬍ .0001, reflecting the fact that the influence of calculation on dwell time was greater when participants had to decide how to exit the targets on the basis of whether the current sum was even or odd than when they did not, either in the thick-or-thin or no-decision exiting conditions. Consistent with this outcome, there was no interaction between target exit group and primary task in an ANOVA limited to the thick-or-thin and no-decision target exit conditions ( p ⬎ .40). Meanwhile, the omnibus ANOVA also revealed a significant interaction between primary task and step number, F(3, 480) ⫽ 25.09, MSE ⫽ 16,832, p ⬍ .0001, such that step number had a more marked effect in the nested task than in the aiming task, as observed in Experiment 1. Movement times. For movement time (Figure 6, Panel C), a Target Exit Group ⫻ Primary Task (nested and aiming) ⫻ Step Number (Steps 3 through 6) ANOVA yielded a significant threeway interaction, F(6, 480) ⫽ 3.10, MSE ⫽ 2,609, p ⬍ .01, such that there were effects of primary task in the no-decision and thick-or-thin groups but not in the even-or-odd group, and the effect of step number was more pronounced in the nested task than in the aiming task, especially for the no-decision and thick-or-thin groups. This interpretation of the three-way interaction was supported by a two-way interaction between target exit group and primary task, F(2, 160) ⫽ 19.83, MSE ⫽ 90,923, p ⬍ .0001, such that movement time was longer in the nested task than in the aiming task for the thick-or-thin group, F(1, 58) ⫽ 113.10, MSE ⫽ 24,535, p ⬍ .0001, and for the no-decision target group, F(1, 58) ⫽ 41.62, MSE ⫽ 209,672, p ⬍ .001, but not for the even-or-odd group ( p ⬎ .60). Moreover, there was a significant two-way interaction between group and step number, F(6, 480) ⫽ 3.27, MSE ⫽ 3,205, p ⬍ .01. Of further interest, movement time differed between groups, F(2, 160) ⫽ 27.53, MSE ⫽ 245,838, p ⬍ .0001, such that movement time was shorter in the no-decision group than in the other two groups. The main effect of task was also significant, F(1, 160) ⫽ 72.99, MSE ⫽ 90,923, p ⬍ .001, with the nested task having longer movement times than the aiming task.

Discussion Experiment 2 was designed to uncover the source of movement lengthening in the nested task of Experiment 1. We considered two hypotheses. The online slowing hypothesis claimed that the slowing was due to concurrent math. The indirect slowing hypothesis claimed that some math was completed before movement initiation occurred and that movement planning suffered as a result of interference, causing movements to be slower than usual. The results of Experiment 2 supported the online slowing hypothesis but not the indirect slowing hypothesis. As predicted by the online slowing hypothesis, movement times were longer in the nested task than in the aiming task for participants in the no-decision group, as found in Experiment 1. Movement times were also longer in the nested task than in the aiming task for participants in

the thick-or-thin target exit condition. This outcome extends the movement-time difference to a new task situation (thick-or-thin target exiting). Most important, however, the movement-time difference between the nested and aiming-only tasks was eliminated in the even-or-odd target exit condition. Here participants were forced to calculate before moving. Elimination of the otherwise robust lengthening of movement time in the context of math when math had to be completed before movement commenced is exactly what was predicted by the online slowing hypothesis. Also in accord with the online slowing hypothesis, dwell times were longer when participants had to complete calculations before moving than when they could move while calculating. Finally, and again consistent with the online slowing view, step times in the no-decision group were virtually identical in the nested and arithmetic-only primary tasks, as in Experiment 1 (for calculation steps). Altogether, it appeared that participants in Experiment 2, like participants in Experiment 1, preferred to off-load their calculation onto movement when they could do so, apparently ending their movements when they finished their calculations.

General Discussion We began this article with a hypothetical story about a student walking between meetings with professors in the Math and Psychology Departments. The student had to coordinate her perceptual-motor activity (walking) with her intellectual activity (preparing to argue the next point in her thesis). We suggested that if the time she needed to prepare her next thesis point happened to be somewhat longer than the time needed to walk between the offices, her optimal scheduling strategy would have been to lengthen her walking time enough to arrive at each professor’s office with the next point worked out. The experiments reported here were designed to test this optimal scheduling prediction. We used a nested aiming–arithmetic task to investigate how manual aiming and mental arithmetic are coordinated when the two types of processes are performed to achieve a common goal. In Experiment 1 we obtained evidence for the strategy attributed to the thoughtful doctoral candidate. Participants appeared to calculate while moving from one target to the next, taking longer to move between the targets when calculations were required than when calculations were not required. Step times for calculation steps in the nested task were about the same as step times for calculation steps in the arithmetic-only task, suggesting that participants tried to arrive at each target when they completed the current calculation. Such synchronization would have avoided dead times when aiming occurred but arithmetic did not. Moreover, by arriving at each target only after the previous calculation was completed, participants would have avoided the situation of having to begin a new calculation when the previous calculation was not yet completed. In Experiment 2 we tested whether this parallel scheduling strategy was in fact more adaptive than a serial strategy. The results confirmed that it was. When participants were forced to use a serial scheduling strategy they performed slowly, but when they were free to use a parallel scheduling strategy they performed as quickly and in essentially the same way, temporally speaking, as did participants in Experiment 1.

COGNITIVE AND PERCEPTUAL-MOTOR SCHEDULING

Questions About the Scheduling Interpretation Obtaining support for the parallel scheduling model raises a number of questions, some of which are discussed below. 1. Under what variations of the present task would the parallel scheduling strategy no longer be used? Variables that should affect the likelihood of using the parallel scheduling strategy are the relative times needed for calculation and for movement. These times were not independently varied in the present experiments but could be in future experiments. So could the difficulty of movement, holding movement time constant. We would expect that a more cognitively demanding movement (e.g., moving through a field of obstacles rather than through an obstacle-free field for which the same time is needed) would lower the likelihood that participants would off-load their math onto their movements. This manipulation would let us evaluate the contributions of interference or resource limitations to scheduling. 2. What internal events trigger movement initiation and movement termination? The present data suggest that participants began to calculate before they started to move and stopped moving when they were done calculating. If the initiation of movement was triggered by completion of some stage of calculation, a manipulation that increases the duration of that stage should selectively elongate dwell time. Similarly, if the termination of movement was triggered by completion of some later stage of calculation (e.g., arriving at the answer), a detailed examination of the kinematics of participants’ movements might reveal this. One might see a shift from slow to quick movements when participants complete the math. Such shifts are not usually observed in visually guided aiming. Instead, one usually sees slowing of the hand as the hand approaches the target. Future research could include precise recordings of the kinematics of the hand. Such recordings were not possible here. 3. Were the start and end of movement triggered by progress in calculation, as assumed in the last paragraph, or were they independently timed? In the latter case (independent timing), if we suddenly changed the difficulty of calculation, we would not expect to see an immediate change in movement timing. In the former case (dependent timing), if we suddenly changed the difficulty of calculation, we would expect immediate movementtiming changes. 4. Was the lengthening of dwell time and movement time actively scheduled, as we have argued, or, contrary to everything we have said, was it just a result of passive interference? We favor the active scheduling account for two main reasons. First, this account predicted that both dwell time and movement time would be stretched to match the time to perform arithmetic. We observed just this pattern (i.e., dwell times and movement times were lengthened when calculation could be done during movement, which in turn caused step times to be statistically indistinguishable from step times in arithmetic-only conditions). Second, as shown in the next section, interference had an observable effect in one condition of Experiment 2, indicating that our procedure allowed us to pick up “pure interference.”

Results Awaiting Explanation Although we have been able to explain most of our results, some other results still await explanation. One is that in Experiment 2,

217

movement times for participants in the nested condition were somewhat longer in the even-or-odd target exit condition (left column of Figure 6, Panel C) than in the no-decision target exit condition (right column of Figure 6, Panel C). This result seems puzzling because, according to our model, participants should off-load calculations onto movement time in the no-decision condition but should not do so in the even-or-odd target exit condition. Indeed, a major source of support for the online slowing model was that we could eliminate the difference between movement times in the nested and aiming-only conditions when participants were forced to calculate before moving (in the even-or-odd target exiting task). If participants calculated while aiming in the nodecision exiting condition but did not calculate while aiming in the even-or-odd exit condition, why were movement times longer when calculations did not co-occur with movement than when they did? This question can be answered by appealing to the kinematics of the movements that were required in the three exit conditions. In the even-or-odd and thick-or-thin exit conditions, the cursor followed a circuitous path to the next target. By contrast, in the no-decision exit condition, the cursor could follow a direct path to the next target. Because longer movement paths take more time than do shorter paths, as implied by Fitts’ Law, it is not surprising that movement times were longer when the cursor had to pass through the top or bottom of the exited target on its way to the next target than when the cursor could move directly to the next target. Inspection of the movement times in Figure 6 turned up another unanticipated outcome, however. In Experiment 2, movement times were longer in the thick-or-thin nested condition (middle column of Figure 6, Panel C) than in the even-or-odd nested condition (left column of Figure 6, Panel C). Why? Both of these conditions required circuitous movement paths, so unless the movement paths were more circuitous in the thick-or-thin nested condition than in the even-or-odd nested condition (which we have no reason to believe was the case), some other factor must have been at play. The most obvious hypothesis is that in the even-orodd condition, participants were forced to perform serially (calculate before aiming), whereas in the thick-or-thin condition they could use parallel processing (calculate while aiming). It is not surprising, then, that movement times were shorter in the even-orodd condition than in the thick-or-thin condition. Indeed, though we did not predict this outcome, it is fitting that we obtained it. A third outcome needing explanation is related to the middle column of Panel B in Figure 6. In Experiment 2, step times in the thick-or-thin condition were longer for the nested task than for the arithmetic-only task. This outcome is surprising because in the nested task of both the thick-or-thin exiting condition and the no-decision exiting condition, participants could adopt a parallel schedule. Why then were step times longer for the nested task than for the arithmetic-only task in the thick-or-thin exiting condition but not in the no-decision condition? We can only speculate on the reason, but we believe this outcome was related to the demands of movement planning or control. In the thick-or-thin nested task, participants had to guide the cursor through a curved path out of the current target and then on to the next target. By contrast, in the no-decision target exit condition, participants merely needed to move the cursor along a straight path to the next target. The demands of controlling the cursor’s circuitous path in the nested thick-or-thin exit condition may have interfered with participants’

SHIN AND ROSENBAUM

218

other cognitive processes. This explanation, though speculative, suggests that interference may have disrupted scheduling.

Conclusion We end this article by noting again that in everyday life, dual tasks often have a nested functional arrangement, yet most research in experimental psychology that has used dual-task methodology has used pairs of tasks that were arbitrarily related. The previous research has revealed that interference and resource limitations play an important role in constraining dual-task performance. On the other hand, the earlier work has generally not permitted precise a priori predictions about the timing and accuracy of any given task when paired with another task. In the present study, we pursued another approach that we thought might allow for more precise predictions. That approach, based on scheduling theory, sees the actor as being capable of timing task activities to optimize the speed and accuracy of their completion. To evaluate the scheduling view, we focused on one pair of functionally related tasks: aiming and mental arithmetic. Consistent with the main prediction of scheduling theory, we found that participants “stretched” their dwell times and movement times so delays between successive target entries (step times) were about the same in the nested and arithmetic-only conditions. The fact that scheduling theory could successfully predict this outcome hints at the power of this little-used approach for investigating the control of cognitive and perceptual-motor processes. In pointing to the potential of this approach, however, we do not mean to discredit interference or resource-limitation views. Indeed, as mentioned in the last section, interference cannot be left out of the scheduling picture. It will be important in future research to see how scheduling, interference, and resource limitations mutually constrain each other. Regardless of how that future work turns out, the findings obtained here make clear that remarkable coordination can be achieved between cognitive and perceptual-motor processes. This outcome shows that the two levels of control are not divided. Instead, they are intimately coupled, so much so that it is difficult to sustain the view that intellectual skills and perceptual-motor skills are different in any meaningful sense. The notion that intellectual skills and perceptual-motor skills share the same psychological underpinnings was recently promoted by Rosenbaum, Carlson, and Gilmore (2001) on the basis of a review of the literature concerning learning in the two domains. The research described in the present article leads to the same notion but through the study of timing.

References Allport, D. A. (1980). Attention and performance. In G. Claxton (Ed.), New directions in cognitive psychology (pp. 112–153). London: Routledge & Kegan Paul. Anderson, D. R., Sweeney, D. J., & Williams, T. A. (1982). An introduction to management science (3rd ed.). St. Paul, MN: West Publishing Co. Baddeley, A. (1986). Working memory. Oxford, England: Clarendon Press/ Oxford University Press. Broadbent, D. E. (1957). A mechanical model for human attention and immediate memory. Psychological Review, 64, 205–215. Charness, N., & Campbell, J. I. D. (1988). Acquiring skill at mental

calculation in adulthood: A task decomposition. Journal of Experimental Psychology: General, 117, 115–129. Ehrenstein, A., Schweickert, R., Choi, S., & Proctor, R. W. (1997). Scheduling processes in working memory: Instructions control the order of memory search and mental arithmetic. Quarterly Journal of Experimental Psychology: Human Experimental Psychology, 50(A), 766 – 802. Elio, R. (1986). Representation of similar well-learned cognitive procedures. Cognitive Science, 10, 41–74. Fitts, P. M. (1954). The information capacity of the human motor system in controlling the amplitude of movement. Journal of Experimental Psychology, 47, 381–391. Gentner, D. R. (1987). Timing of skilled motor performance: Tests of the proportional duration model. Psychological Review, 94, 255–276. Graham, R. L. (1978). The combinatorial mathematics of scheduling. Scientific American, 238, 124 –132. Greenwald, A. G., & Shulman, H. G. (1973). On doing two things at once: II. Elimination of the psychological refractory period effect. Journal of Experimental Psychology, 101, 70 –76. Healy, A. G., & Nairne, J. S. (1985). Short-term memory processes in counting. Cognitive Psychology, 17, 417– 444. Heuer, H., & Wing, A. M. (1984). Doing two things at once: Process limitations and interactions. In M. M. Smyth & A. M. Wing (Eds.), The psychology of human movement (pp. 183–213). London: Academic Press. Hirst, W., & Kalmar, D. (1987). Characterizing attentional resources. Journal of Experimental Psychology: General, 116, 68 – 81. Kahneman, D. (1973). Attention and effort. Englewood Cliffs, NJ: Prentice Hall. Kantowitz, B. H. (1974). Double stimulation. In B. H. Kantowitz (Ed.), Human information processing: Tutorials in performance and cognition (pp. 83–131). Hillsdale, NJ: Erlbaum. Keele, S. W. (1973). Attention and human performance. Pacific Palisades, CA: Goodyear. Kerr, B. (1973). Processing demands during mental operations. Memory & Cognition, 1, 401– 412. MacKenzie, I. S. (1992). Fitts’ Law as a research and design tool in human– computer interaction. Human-Computer Interaction, 7, 91–139. McLeod, P. (1980). What can probe RT tell us about the attentional demands of movement? In G. E. Stelmach & J. Requin (Eds.), Tutorials in motor behavior (pp. 579 –589). Amsterdam: North-Holland. Meyer, D. E., & Kieras, D. E. (1997a). A computational theory of executive cognitive processes and multiple-task performance: Part 1. Basic mechanisms. Psychological Review, 104, 3– 65. Meyer, D. E., & Kieras, D. E. (1997b). A computational theory of executive cognitive processes and multiple-task performance: Part 2. Accounts of psychological refractory-period phenomena. Psychological Review, 104, 749 –791. Meyer, D. E., Smith, J. E. K., Kornblum, S., Abrams, R. A., & Wright, C. E. (1990). Speed–accuracy tradeoffs in aimed movements: Toward a theory of rapid voluntary action. In M. Jeannerod (Ed.), Attention and performance XIII: Motor representation and control (pp. 173–226). Hillsdale, NJ: Erlbaum. Netick, A., & Klapp, S. T. (1994). Hesitations in manual tracking: A single-channel limit in response programming. Journal of Experimental Psychology: Human Perception and Performance, 20, 766 –782. Norman, D. A., & Shallice, T. (1980). Attention to action: Willed and automatic control of behaviour (CHIP Report No. 99). San Diego: University of California, San Diego. Ogden, G. D., Levine, J. M., & Eisner, E. J. (1979). Measurement of workload by secondary tasks. Human Factors, 21, 529 –548. Pashler, H. (1984). Processing stages in overlapping tasks: Evidence for a central bottleneck. Journal of Experimental Psychology: Human Perception and Performance, 10, 358 –377. Rosenbaum, D. A., Carlson, R. A., & Gilmore, R. O. (2001). Acquisition

COGNITIVE AND PERCEPTUAL-MOTOR SCHEDULING of intellectual and perceptual-motor skills. Annual Review of Psychology, 52, 453– 470. Rosenbaum, D. A., Vaughan, J., Barnes, H. J., & Jorgensen, M. J. (1992). Time course of movement planning: Selection of hand grips for object manipulation. Journal of Experimental Psychology: Learning, Memory, and Cognition, 18, 1058 –1073. Schweickert, R. (1980, August 8). Critical-path scheduling of mental processes in a dual task. Science, 209, 704 –706. Sternberg, S. (1969). The discovery of processing stages: Extensions of Donders’ method. In W. G. Koster (Ed.), Attention and performance II (pp. 276 –315). Amsterdam: North-Holland. Van Galen, G. P., Smyth, M. M., Meulenbroek, R. G. J., & Hylkema, H. (1989). The role of short-term memory and the motor buffer in handwriting under visual and non-visual guidance. In R. Plamondon, C. Y. Suen, & M. L. Simner (Eds.), Computer recognition and human production of handwriting (pp. 253–271). Singapore: World Scientific Publishing Company. Welford, A. T. (1952). The “psychological refractory period” and the

219

timing of high-speed performance: A review and a theory. British Journal of Psychology, 43, 2–19. Wenger, J. L., & Carlson, R. A. (1995). Learning and coordination of sequential information. Journal of Experimental Psychology: Human Perception and Performance, 21, 170 –182. Wickens, C. D. (1980). The structure of attentional resources. In R. S. Nickerson (Ed.), Attention and performance VIII (pp. 239 –257). Hillsdale, NJ: Erlbaum. Wickens, C. D., & Liu, Y. (1988). Codes and modalities in multiple resources: A success and a qualification. Human Factors, 30, 599 – 616. Yee, P. L., Hunt, E., & Pellegrino, J. W. (1991). Coordinating cognitive information: Task effects and individual differences in integrating information from several sources. Cognitive Psychology, 23, 615– 680.

Received April 7, 1999 Revision received November 13, 2001 Accepted November 13, 2001 䡲