How Planful Is Routine Behavior? A Selective-Attention ...ema/sam/EGPatsenko.pdf · How Planful Is...

How Planful Is Routine Behavior? A Selective-Attention Model ofPerformance in the Tower of Hanoi

Elena G. Patsenko and Erik M. AltmannMichigan State University

Routine human behavior has often been attributed to plans—mental representations of sequences goalsand actions—but can also be attributed to more opportunistic interactions of mind and a structuredenvironment. This study asks whether performance on a task traditionally analyzed in terms of plans canbe better understood from a “situated” (or “embodied”) perspective. A saccade-contingent display-updating paradigm is used to change the environment by adding, deleting, and moving task-relevantobjects without participants’ direct awareness. Response latencies, action patterns, and eye movementsall indicate that performance is guided not by plans stored in memory but by a control routine bound toobjects as needed by perception and selective attention. The results have implications for interpretingeveryday task performance and particular neuropsychological deficits.

Keywords: selective attention, eye movements, Tower of Hanoi, situated cognition, problem solving

Much of our life’s activity consists of relatively orderly routines,such as getting ready in the morning, preparing breakfast, driving,cleaning the house, etc. Such routines represent skilled behavior inthe sense that they have been shaped by repeated performance ofthe task but may still involve parameters that change from oneinstance to the next. In terms of Anderson’s (1995) theory oflearning, we would say they correspond to the associative stage, inwhich one operation has come to fairly directly trigger the next.Early approaches to analyzing such behavior focused on the pos-sibility that it was guided by rich internal representations, such asplans specifying a series of goals to achieve or actions to take (e.g.,Ernst & Newell, 1969; Miller, Galanter, & Pribram, 1960; Newell& Simon, 1972), and less on the possibility that it was guided inlarge part by processes like perception and attention that are lesscentrally cognitive. This emphasis on internal representations mayhave been a reaction to the neglect of such constructs during thebehaviorist era but was in any case propelled by the digital com-puter as a metaphor for human information processing, whichallowed mental representations and processes to be codified inprecise terms and then simulated. Plans in particular could berepresented as programs and then executed to prove their func-tional sufficiency to accomplish some complex task. Such dem-onstrations often made use of puzzle tasks that, like the Tower ofHanoi, require some kind of mental search through a space ofpossible solution paths, the results of which are then useful to storein memory to guide actual task performance. Such tasks also often

have a simple apparatus that does not change dynamically on itsown, such that adequate “world models” can be built directly intothe system, circumventing the need to model processes like per-ception and attention that would ordinarily construct representa-tions of the world from perceptual inputs.

Under the influence of the computer metaphor, the plan-basedapproach to analyzing routine behavior may have overestimatedthe power of central cognitive processes, while at the same timeunderestimating the power of peripheral processes to guide behav-ior using information extracted from the environment. Such is theview of the situated (or embedded) approach that evolved inresponse, which proposes that routine or otherwise orderly behav-ior can emerge from interactions between organism and environ-ment that foster opportunism at the process level (Agre & Chap-man, 1987; Ballard, Hayhoe, Pook, & Rao, 1997; Clark, 1997;Glenberg, 1997; Gray, Sims, Fu, & Schoelles, 2006; Suchman,1987). For example, the behavior of a robot driving to a table topick up a can of soda can be characterized in terms of a drivingprocess producing a physical proximity to the table, with thisproximity in turn triggering a reach for the can (Brooks, 1991). Inother words, although the driving and reaching processes in somesense communicate, they do so indirectly through the environment,rather than directly through internal channels using shared internalrepresentations.

The situated approach also proposes that internal representationsare generally simple and sparse. For example, Agre and Chapman(1987) developed a software agent that played the arcade gamePengi, controlling a virtual penguin whose task was to chase andattack virtual bees in a maze of obstacles. The penguin’s internalrepresentations were “indexical,” meaning they represented ob-jects in relation to the agent’s body, and “functional,” meaning thatthey pointed only to elements that were directly relevant to theagent’s activity. For example, a “bee-that-I’m-attacking” index,which was kept locked to a particular bee by the perceptual system,identified the current object of the penguin’s activity. A few suchrepresentations, carefully chosen, allowed the penguin to exhibit

Elena G. Patsenko and Erik M. Altmann, Department of Psychology,Michigan State University.

This research was supported by Office of Naval Research GrantsN00014-06-1-0077 and N00014-09-10093 to Erik M. Altmann. The au-thors thank John R. Anderson for his comments and suggestions forimprovement.

Correspondence concerning this article should be addressed to Elena G.Patsenko, Department of Psychology, Michigan State University, EastLansing, MI 48824. E-mail: [email protected]

Journal of Experimental Psychology: General © 2010 American Psychological Association2010, Vol. 139, No. 1, 95–116 0096-3445/10/$12.00 DOI: 10.1037/a0018268

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what seemed like goal-directed behavior without much of a worldmodel and without any kind of planning system.

The situated approach has not yet been applied to the kinds oftasks for which the plan-based approach seems ideally suited:those with complex rules that seem to require searching a space ofsolution paths, but with a simple and static environment, such thatplans encoded during the search are useful for guiding behaviorduring actual performance. In this study, we ask whether thesituated approach can actually improve on plan-based models of(human) performance in one canonical planning task, the Tower ofHanoi puzzle. If so, this might indicate a need to reevaluate themechanisms of “planful” behavior more broadly.

The article is organized as follows. In the next section, wedescribe the puzzle and the two most current cognitive models,both of which assume that behavior is guided by some kind of planstored in memory. We then describe our own selective-attentionmodel and its perceptual and attentional mechanisms (a computa-tional simulation of which is presented in the Appendix). We thenpresent three experiments that test contrasting predictions of theplan-based and selective-attention models, using a saccade-contingent display-updating procedure that allows us to change theworld—meaning, here, the puzzle display—while the system isfunctionally blind and thereby probe the nature of the representa-tions guiding behavior. In the General Discussion, we reassessevidence previously offered in support of a plan-based account ofTower of Hanoi performance and examine implications of ourwork for interpreting everyday tasks as well as neuropsychologicalimpairments assessed with this and similar puzzle tasks.

The Tower of Hanoi Task and Current Models

A sample Tower of Hanoi problem is shown in Figure 1a. Thereare three pegs holding sets of disks of different sizes. In the initialstate of the puzzle, the disks are typically arranged as here in a

pyramid on one of the pegs, and in the final state, the disks aretypically arranged in a pyramid on a different peg. The task is totransform the initial state to the final state by moving disks frompeg to peg, subject to the rules that (a) only one disk can be movedat a time, (b) a larger disk cannot be placed on a smaller one, and(c) a disk can only be moved if it is the top disk on a peg.

One way to approach solving the puzzle is to use the goal-recursive algorithm for planning a given move. For the problemstate shown in Figure 1a, this algorithm would apply as follows.To move the tower from Peg A to Peg C (the overall goal), thelargest disk must be moved to Peg C, but this move cannot bemade yet because it is “blocked” by the other disks sitting on topof the largest disk. Thus, the goal to move the largest disk must besuspended. The system then recurses, setting itself the goal tomove the second-largest disk out of the way to Peg B to allow thelargest disk to be moved. The second-largest disk also cannot bemoved yet, so the system recurses again to the next “blocking”disk, and so on, until it reaches the smallest disk on the tower andmoves that to Peg C, which at that point represents “out of theway” for moving the second-smallest disk. There are other ways tosolve the puzzle (e.g., Simon, 1975), and studies of how peopleacquire strategies to solve this task leave participants to discoveran algorithm on their own (e.g., Anzai & Simon, 1979). However,in studies that examine performance on this task once it hasbecome routinized, the goal-recursive algorithm is often simplytaught to participants to reduce strategic variability (e.g., Ander-son, Kushmerick, & Lebiere, 1993), which is also the approach wetake here.

Current models of routinized performance on this task (Altmann& Trafton, 2002; Anderson & Douglass, 2001) assign two distinctfunctional roles to the goal-recursive algorithm. The first is tocompute where to move the disk at the top of a tower, as describedabove. The second is to encode a plan in memory representing the

Figure 1. The goal-recursive algorithm in the Tower of Hanoi puzzle. a: The rule applied to the initial state ofa five-disk problem. The final state is for the tower to be on Peg C. The arrows indicate intended diskmovements. Arrows, disk labels, and peg labels were not on the problem display shown to participants. b:Example saccadic eye movements during application of the rule. Arrows indicate saccades and dots indicatefixations and were, again, not shown to participants. c: The mental representation generated by the goal-recursivealgorithm according to the ordered-goals model, at the point where the recursion has reached Disk 1. Goals areordered by recency, with the current goal on top and the oldest goal on the bottom.

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sequence of goals generated by the algorithm as it executes. Thisplan then guides processing on later trials. Figure 1c shows theplan encoded by the goal-recursive algorithm as it applies inFigure 1a. The plan would then guide subsequent moves as fol-lows: Once the first move is made (Disk 1 to Peg C), the suspendedgoal to move Disk 2 to Peg B would be retrieved, and this movewould then be made, because it can be at that point. After that, thesuspended goal to move Disk 3 to Peg C would be retrieved. Thisgoal cannot be achieved at this point (because Disk 1 is on Peg C),but it still guides behavior in that it triggers a new application ofthe goal-recursive algorithm, which starts with the goal to moveDisk 3 to Peg C (and generates, as the next move at that point, Disk1 from Peg C to Peg B). The number of goals stored in the planthus fluctuates, with goals removed as they are achieved and addedwhenever the goal-recursive algorithm executes; the maximumnumber of goals in the plan at any point corresponds to the numberof disks in the problem.

The two plan-based models we examine here both assume thatgoals are encoded in memory and retrieved to guide behavior butdiffer in their assumptions about how information about goalsequencing is represented. What we refer to as the ordered-goalsmodel of Anderson and Douglass (2001) assumes that goals areordered internally in a sequential representation that correspondsto the order in which they need to be retrieved to guide behavioralong the optimal solution path. In contrast, what we refer to as theprimed-goals model of Altmann and Trafton (2002) assumes thatthe retrieval cue for a goal is the disk coded in that goal and thatheuristics guide the system to attend to the correct disk in thepuzzle display at the correct time to cue retrieval of the correctgoal. The primed-goals model constitutes a shift from internal toexternal representations of goal-sequencing information and thus astep on the path to the hypothesis of the present study but preservesthe key assumption that goals are encoded in memory by thegoal-recursive algorithm and then retrieved later to guide behavior.Thus, both the ordered-goals and primed-goals models assume thatperformance on this task is “planful,” meaning guided by plansstored explicitly in memory.

In the selective-attention model we discuss next, we assign onlyone functional role to the goal-recursive algorithm, which is tocompute where to move the disk at the top of the tower; thealgorithm encodes no plan in memory to guide future processing.

The Selective-Attention Model

We assume that routinized performance on the Tower ofHanoi—after initial learning, once people have basically figuredout how to solve each new problem they are given—draws less onmemory representations and more on perceptual and attentionaloperations than the plan-based models we discussed above. Themain perceptual operation is to recognize a tower of disks when ithas been newly assembled. This new tower is then bound to anattentional pointer or index that functions like the FINST constructproposed by Pylyshyn and colleagues (e.g., Pylyshyn, 2000). AFINST binds a parameter of a conceptual representation to anobject in the environment. Results of multiple-object trackingstudies suggest that there are four or five FINSTs available forconcurrent and independent use. In our case, the relevant concep-tual representation is an abstract control routine that governsperformance on the Tower of Hanoi task. The routine is abstract in

that it has parameters that have to be bound to specific objects forit to execute. When it does execute, it directs actions that changethe world so as to trigger a new round of parameter bindings andthus another round of execution. The routine is triggered byrecognition of a new tower, then takes another step in disassem-bling an old tower, then invokes the goal-recursive algorithm onthe new tower to compute the next move. The basic memoryrepresentations from which the model draws are thus attentionalindexes that track external objects: a newly recognized tower, anold tower that is being disassembled, and a disk displaced from thelatter to which to transfer the former. We elaborate on thesemechanisms in the following sections and in the Appendix.

Recognizing a Tower

We assume that, after participants have learned to perform thetask, they are able to recognize towers of disks as objects relevantto performance. There is behavioral evidence for such tower rec-ognition (Ruiz & Newell, 1989), and our experience with the tasksuggests that towers are highly salient, given that performanceinvolves continually assembling new towers in the course ofdisassembling old ones.

We define a tower operationally as a set of two or more disks ona peg in which the smallest is Disk 1 and in which disks differ byonly one step in size. Figure 2 shows two different towers thatmeet this definition. In Figure 2a, the tower on Peg C comprises allthree disks. In Figure 2b, the tower on Peg C comprises Disks 1and 2 only, given that Disks 2 and 5 differ by more than one stepin size; this jump breaks continuity in the Gestalt sense, which wetake as a perceptual basis for the system to limit the extent of therecognized tower. We take tower recognition to be a bottom-upperceptual operation triggered whenever a tower is completed—that is, when Disk 1 is placed on top.

The new tower is relevant because it becomes the input to thegoal-recursive algorithm, which computes how to move it toanother peg. The system therefore points to the new tower with anew-tower index, which is updated when a new tower is recog-nized. In terms of the arcade-game example we mentioned earlier(Agre & Chapman, 1987), updating the new-tower index is similarto updating the penguin’s “bee-that-I’m-attacking” index to pointto a new bee after the old bee has been successfully attacked.Unlike the old bee, however, the old tower may still be relevant, ifit has not yet been fully disassembled. Therefore, the system alsopoints to the old tower with an old-tower index.

The Recognize-Displace-Compute Routine

The assembly and disassembly of towers of disks are relatedfunctionally in a way that causes them to be interleaved behavior-ally, and we assume that after initial learning, the system hasacquired a control routine that manages this interleaving in con-junction with environmental triggers. Specifically, we assume thatrecognizing a new tower triggers a move to “displace” another diskfrom a tower that is being disassembled and that this displacementtriggers a new application of the goal-recursive algorithm. Thealgorithm computes the correct move for Disk 1, given the goal totransfer the new tower as a whole to the displaced disk. Infunctional terms, what this routine accomplishes is to displace adisk from the old tower and add it to the new tower from under-

97SELECTIVE ATTENTION IN ROUTINE BEHAVIOR

neath, in a sense, which then opens up another peg to displace yetanother disk from the old tower and thus make progress towardfully disassembling the old tower as part of reconstructing it onanother peg. (We make no assumptions about whether participantsrepresent the functional effect of this routine; they could also learnit simply by generalizing over action sequences during initialexperience with the task.) The displaced disk plays a central rolein this sequence, and to represent this role, the system points to itwith a just-displaced-disk index.

To illustrate these various constructs, Figure 2a shows a newtower, signified by the box, and a just-displaced disk, signified bythe oval. In the state one move prior to the state shown there, Disk4 was on Peg A, Peg B was empty, and Disk 1 had just been movedto Peg C, completing the new tower. Disk 4 was then displacedfrom Peg A to Peg B to become the just-displaced disk. The nextstep is for the goal-recursive algorithm to compute the correct nextmove for Disk 1, which is to move it from Peg C to Peg B.

To illustrate by analogy to another situated system, earlier, wementioned Brooks’s (1991) example of a robot that drives towarda table, at which point the proximity of the table triggers a reachfor the soda can on the table. The role of proximity in triggeringthe reach there is analogous to the role of recognition of a newtower triggering further disassembly of the old tower. In bothcases, control passes from one procedural step to another throughthe environment, with the first step changing the world in a waythat triggers the second.

An incidental assumption not central to our hypothesis (see theAppendix) is that after the move indicated by recognize-displace-compute, the next two moves are guided by heuristics that we carryover from previous models. The first moves Disk 2 to the other peg(an instantiation of the “don’t-undo” heuristic; Altmann & Trafton,2002; VanLehn, 1991), and the second moves Disk 1 on top ofDisk 2 (the “1-follows-2” heuristic of Altmann & Trafton, 2002).This three-move sequence (which Karat, 1982, interpreted basi-cally as a procedural chunk) transfers Disks 1 and 2 of the newtower (which may be the whole tower) to another peg. For exam-ple, in Figure 2a, Disks 1 and 2 would be transferred from Peg Cto Peg A.

The Empty Peg

We assume that the perceptual and attentional operations de-scribed above determine what task-relevant objects are selected forattention during routine performance on this task, implying thatother objects on the puzzle display are not selected for attention,including, specifically, large disks that have not been processedrecently. When such a disk is the top disk on a peg, we take it tobe ignored, as if the peg is physically empty (holds no disks at all).An empty peg plays an important role in performance, in that it isthe destination for a disk being displaced from the old tower.

The state in Figure 2a resulted from displacing Disk 4 to Peg B,which at that point was physically empty. In contrast, the state inFigure 2b resulted from displacing Disk 3 to Peg A, which at thatpoint held Disk 6, but Disk 6 was not part of the new tower (Disks1 and 2 on Peg C) or the old tower being disassembled at that point(Disks 3 and 4 on Peg B), so was not selected for attention. Thus,Peg A was effectively empty.

Overview of Experiments

In three experiments, we tested whether performance on theTower of Hanoi is guided by plans—sets of move-specific goalsformulated ahead of time, stored in memory, and retrieved toguide behavior— or by an abstract control routine bound op-portunistically to specific objects in the environment. The basicmethod was to alter the puzzle display “invisibly” during per-formance such that information extracted from the changeddisplay would lead to one behavior, whereas a goal retrievedfrom memory—and encoded from the prechange display—would lead to a different behavior. We changed the displayduring saccades, when the visual system is functionally blind(e.g., Li & Matin, 1997; O’Regan, 1992), so as to mask thevisual transients that would ordinarily trigger direct awarenessof environmental changes.

In all three experiments, participants were shown the goal-recursive algorithm and allowed to practice with it, then performedthree target problems for which we collected data. The first andthird target problems were control problems, and the second was

Figure 2. Problem states used to illustrate the recognize-displace-compute routine. a: Two moves ago, thesystem placed Disk 1 on Peg C to complete a new tower (marked by the box). One move ago, the systemdisplaced Disk 4 from the old tower on Peg A, which is currently being disassembled, to the empty Peg B. Thegoal-recursive algorithm starts with the goal to move Disk 3 of the new tower to the just-displaced disk (markedby the oval) and identifies the correct next move as Disk 1 to Peg B. b: Two moves ago, the system placed Disk1 on Peg C to complete a new tower (marked by the box). One move ago, the system displaced Disk 3 from theold tower on Peg B, which is currently being disassembled, to the subjectively empty Peg A. The goal-recursivealgorithm starts with the goal to move Disk 2 of the new tower to the just-displaced disk (marked by the oval)and identifies the correct next move as Disk 1 to Peg B.


the critical problem, in which we made saccade-contingent displaychanges. In all three problems, the disks were arranged in a singletower on one peg in the initial state and on a different peg in thefinal state, with the initial and final pegs differing between prob-lems.

In Experiment 1, the changes in the critical problem involvedadding a disk to the initial tower and then exchanging disks laterin the problem, such that the participant started a five-disk problemand finished a six-disk problem with the same final state. InExperiment 2, the change was to delete two disks from the initialtower in the critical problem, such that the participant started aseven-disk problem and finished a five-disk problem. In Experi-ment 3, the change was to perform the correct move for theparticipant at a particular intermediate state of the critical problem.In each case, we created a “treadmill” of routine behavior andmanipulated the environment to see what was guiding it: goalsstored in memory representing the old environment, or a controlroutine bound as needed with details from the current environment.

Experiment 1: Disk Addition Followed by DiskExchange

In Experiment 1, the initial state of the critical problem con-tained five disks, as in the sample problem shown in Figure 1.Between Moves 2 and 3 we added a disk to the top of the initialtower. Figure 3 shows the problem state before and after theaddition (see Figures 3a and 3b, respectively). The new disk,labeled 3a (although not on the participant’s display), was added asthe participant made a saccade from Disk 2 to Disk 1 to pick upDisk 1 and place it on Disk 2. To preserve uniform size differences

between all disks, Disk 3a was made the original size of Disk 2,and Disks 1 and 2 were slightly shrunk. The visual transientsresulting from all these changes occurred during a saccade, soshould not have been perceived.

Later in the problem, between Moves 10 and 11, we exchangedthe positions of Disks 3 and 3a. The aim was to preserve theoriginal final state of the problem, which we thought participantsmight remember. Figure 3 shows the problem state before and afterthe exchange (see Figures 3c and 3d, respectively). The exchangeoccurred as the participant made the saccade from Disk 2 to Disk1 to pick up Disk 1 and place it on Disk 2. Perceptually, the effectof the exchange is subtle, but its effect is to reverse the directionof Move 12, as we describe in the following.

Diagnostic Moves

We aim to discriminate empirically between the ordered-goals,primed-goals, and selective-attention models by comparing theirpredictions for which moves should show increased responselatencies in the critical problem compared with the control prob-lems. The order-goals model predicts increases on Moves 12, 16,and possibly 4; the primed-goals model predicts an increase onMove 4; and the selective-attention model predicts an increase onMove 12. In the following, we consider Moves 4, 12, and 16 indetail.

Move 4. We illustrate Move 4 of the critical problem incontext of Figure 3. The problem state shown in Figure 3b is theone immediately preceding Move 3, which transfers Disk 1 fromPeg C to Peg B, creating a two-disk tower on Peg B and leaving

Figure 3. Saccade-contingent display changes in the critical problem of Experiment 1. In this instance of thecritical problem, the initial tower was on Peg A, and the instruction was to move it from there to Peg C. a: Theproblem state after Move 2, which is 2AB (transfer Disk 2 from Peg A to Peg B). The display changes that followare to add Disk 3a and shrink Disks 1 and 2. b: The problem state immediately after these changes. Move 3 is1CB, and Move 4 is 3aAC. c: The problem state after Move 10, which is 2CA. The display change that followsis to exchange Disks 3 and 3a. d: The problem state immediately after these changes. Move 11 is 1BA, and Move12 is 3aBC.


Peg C empty. Move 4 is then 3aAC—that is, it transfers Disk 3afrom Peg A to Peg C.

To derive predictions for the ordered-goals model, we needed toask what information might be coded in goals. Anderson andDouglass (2001) did not commit to specific details at this level,which were not relevant to their behavioral tests, so their model is,for our purposes, incomplete. Our approach to filling in the miss-ing details was to assume two different sets that seemed reasonableand test the implications of each. We cannot rule out the possibilitythat other reasonable assumptions make different predictions.

One possibility is that a goal codes the to-be-moved disk and itsdestination peg. This disk-to-peg coding predicts a latency increaseon Move 4 in the critical problem (relative to Move 4 in the controlproblem). The current goal at this point is 3C—that is, to moveDisk 3 to Peg C. After the display change, this goal cannotbe directly achieved, because Disk 3 is now covered by Disk 3a(see Figure 3b). Thus, latency on Move 4 should reflect time forthe system to compute how to achieve 3C. In the control problem,in contrast, 3C can be directly achieved at this point, because Disk3 is the top disk on Peg A (see Figure 3a).

The other possibility is that a goal codes the source and desti-nation pegs for a move. This peg-to-peg coding predicts no latencyincrease on Move 4. The current goal at this point is AC—that is,to move the top disk on Peg A to Peg C. With this goal, the systemcan simply move Disk 3a as it would have moved Disk 3. Wefocus on the peg-to-peg coding on later diagnostic moves.

The primed-goals model also predicts a latency increase onMove 4, because any goal retrieved at this point should not bedirectly achievable. According to this model, the system shouldfocus on the most recently uncovered disk at this point, using thedisk itself as a cue to retrieve a goal indicating where that diskshould be moved. Disk 3 was most recently uncovered, duringMove 2. However, Disk 3 is now covered again, by Disk 3a, so ifthe retrieved goal is relevant to Disk 3, response latency shouldreflect time to compute (using goal-recursion) how to move Disk3a out of the way. If, instead, the system simply focuses on the topdisk of the relevant tower and uses that as the retrieval cue, itshould focus on Disk 3a—which is identical to the original Disk 2,so should retrieve 2B, the original goal for Disk 2. This againcannot be directly achieved, because Peg B holds smaller disks, soresponse latency should again reflect goal-recursion time. Alter-natively, if the system were to detect that 2B had been retrievedonce before and achieved, this should trigger time-consuming

processing that would not be triggered on this move in the controlproblems. Thus, various interpretations of the primed-goals modelall predict a latency increase on Move 4.

The selective-attention model predicts no latency increase onMove 4, because there is no conflict between the external andinternal problem representations. There is now a new two-disktower on Peg B, which was created by Move 3, and recognition ofthis new tower should lead the system to displace another diskfrom the old tower, which is on Peg A, to the empty peg.

Move 12. We illustrate Move 12 of the critical problem incontext of Figure 3d. The state shown there is between Moves 10and 11, immediately after the saccade-contingent exchange ofDisks 3 and 3a. Move 11 is 1BA, and Move 12 is 3aBC.

The peg-to-peg ordered-goals model predicts a latency increaseon Move 12 in the critical problem, because the goal at that pointis not achievable. The goal at that point, which would have beenformulated during goal-recursion on Move 9 before the exchangeof Disks 3 and 3a, is CB. Now, after the exchange, CB cannot bedirectly achieved, because the disk on Peg C is larger than the diskon Peg B. Therefore, this model suggests that we may see illegalmove attempts at this point, if the system does not represent disksizes, but in any case we should see a latency increase as thesystem contends with the fact that it cannot achieve its currentgoal.

The selective-attention model also predicts a latency increase onMove 12. Move 11 completes a new two-disk tower on Peg A,which the system should recognize, triggering a displace operationthat moves a disk from the old tower, which is on Peg C, to Peg B.After the exchange, this move is illegal, so Move 12 should reflecttime to contend with this inconsistency.

Move 16. The problem states immediately prior to Move 16,which vary by problem type, are shown in Figure 4. The state inthe critical problem is shown in Figure 4a, and the state in thecontrol problem is shown in Figure 4b.

The peg-to-peg form of the ordered-goals model, under a keyassumption, predicts a latency increase on this move in the criticalproblem because the current goal is not achievable. The assump-tion, which we address in more detail in the Discussion, is that thesystem is following the plan it formulated earlier, before we addedDisk 3a. Under this assumption, the goal at this point is AC; thiswould have been formulated during goal-recursion on Move 1. Inthe critical problem, AC cannot be directly achieved, because PegC holds smaller disks that are blocking it. Thus, move latency in

Figure 4. The problem states prior to Move 16 in Experiment 1 in (a) the critical problem and (b) the controlproblem. a: The large arrow indicates the move predicted by the peg-to-peg form of the ordered-goals model,and the small arrow indicates the correct move. b: The one arrow indicates the move predicted by both theordered-goals model and the selective-attention model.


the critical problem should reflect time to compute how to movethis tower. In the control problem, AC can be directly achieved,because Peg C is empty.

The selective-attention model predicts no latency increase in thecritical problem, because there is no conflict between the externaland internal problem representations. The previous move com-pleted a new tower, on Peg C in the critical problem and on Peg Bin the control problem, which should lead the system to displaceanother disk from the old tower, which is on Peg A in bothproblems (the old tower consists of Disks 4 and 5 in the criticalproblem and Disks 5 and 6 in the control problem). The systemshould attend to the top disk on Peg A and move it to the emptypeg (Peg B in the critical problem and Peg C in the controlproblem).

Summary. An effect of problem type (critical, control) onresponse latency for Move 12, but not on response latency forMoves 4 and 16, would support the selective-attention model, andwould be evidence against both forms of the ordered-goals modeland the primed-goals model.

Diagnostic Eye Movements

The selective-attention and plan-based models make differentpredictions for where participants should look in particular prob-lem states, on the assumption that people generally look at objectsthey are processing (e.g., Rayner & Liversedge, 2004). Figure 5shows a sample state (the same in both panels) and predicted eyemovements for the two models. The state is the one produced byMove 4AB, which is Move 8 in a six-disk problem in which alldisks are on Peg A in the initial state and Peg B in the final state.According to the selective-attention model, Disk 4 is the just-displaced-disk, Peg A holds the old tower, and Peg C holds thenew tower. The system should next apply the goal-recursive algo-rithm to the new tower, so should fixate next in the vicinity of PegC. According to the plan-based models, Move 4AB achieved agoal, and when a goal is achieved the next operation is to retrievethe next goal from the plan to see if that is also achievable (and, ifnot, compute how to achieve it). The next goal is 5C, or AC inpeg-to-peg format, so the system should fixate next in the vicinityof Peg A. For the primed-goals model, this is a strong prediction,because attending to Disk 5 is necessary to prime retrieval of the

associated goal. In the ordered-goals model, the prediction isweaker in that it rests on the assumption we noted earlier thatpeople generally look at objects they are processing.

Three states in each problem (both control and critical) weregood candidates for this analysis, namely those following Moves4, 8, and 16. In each case, the hypothetical new tower rests on adifferent peg than the disk associated with the hypothetical to-be-retrieved goal, so fixations could be reliably coded as supportingone model or the other. The relevant fixations were coded byassessing for each one whether it fell in the region occupied by apeg.

Method

Participants. Thirty-eight undergraduate students from theMichigan State University psychology subject pool participated inexchange for course credit. All participants had normal orcorrected-to-normal vision.

Given the small number of observations per participant—oneper move for the critical problem, two per move for the controlproblems—we handled errors by excluding participants. An errorwas a move that deviated from the optimal solution path. Aparticipant was included if he or she made no errors on diagnosticmoves in the control problems and no more than two errors on thecombined set of 58 nondiagnostic moves from the first 32 movesof the control problems (these 32 moves are the ones we examinein the Results and Discussion section). Twenty-two participantsmet this accuracy criterion.

Procedure. Participants were tested individually in sessionslasting about an hour. The participant first read a description of therules governing moves in the Tower of Hanoi puzzle: that only onedisk could be moved at a time, that a disk could not be placed ona disk that was smaller than it, and that a disk could not be movedif there was another disk on top of it. The participant was thengiven a four-disk problem and asked to explore it. The participantwas allowed to finish it on his or her own if he or she could; if theparticipant became stuck the experimenter intervened to help. Theexperimenter then presented a new five-disk problem and demon-strated the correct move sequence to solve it, describing thegoal-recursive algorithm in the process. The goal-recursive algo-

Figure 5. Predicted eye movements for the selective-attention and plan-based models. In both panels (whichshow the same state), the previous move was 4AB (Disk 4 moved from Peg A to Peg B). a: Under theselective-attention model, Disk 4 is the just-displaced disk, and the newly assembled tower is on Peg C, so theeyes should move next to the vicinity of Peg C. b: Under plan-based models, 4AB achieves a goal, and the nextgoal is 5C or AC, so the eyes should move next to the vicinity of Peg A.


rithm was described in the following terms (gestures are noted inbrackets):

If you need to move the pyramid to Peg C, then the largest disk [pointsto Disk 5] should go there first. So formulate the goal to move thisdisk onto this peg [points to Peg C]. At the moment, the goal cannotbe achieved, because this disk is blocked by other disks [circles theblocking disks, which are Disks 1, 2, 3, and 4]. Find the largestblocking disk, which is Disk 4, and formulate the goal to move thisdisk out of the way to the peg that differs from the destination peg ofthe largest disk [points to Disk 5]. Check if you can execute the move.Repeat these steps until you find a move that can be executed.

The participant was then given four practice problems, consist-ing of two six-disk problems, a five-disk problem, and anothersix-disk problem. The pegs for the initial and final towers variedacross problems. The experimenter provided help as needed duringthis phase.

The three target problems were then presented: the first controlproblem (six disks), the critical problem in which saccade-contingent display changes were made (initially five disks), andthe second control problem (six disks). The pegs for the initial andfinal towers differed across problems, with the order counterbal-anced across participants. The location for the final tower wasindicated at the start of a problem with an instruction to “Pleasetransfer the tower onto the third peg,” for example. The experi-menter intervened to correct errors during this phase.

Participants moved a disk by clicking it with the mouse, drag-ging it over the destination peg, and releasing the mouse, at whichpoint the disk fell into place. Response latency for a move wastimed from release of the previous disk (or, for Move 1, from theonset of the problem) to release of the disk currently being moved.If the participant attempted an illegal move by trying to place alarger disk on to a smaller disk, the larger disk dropped onto thepeg, but fell to the lowest level on the peg, overlapping withthe disk that was already there. If an illegal move was made duringthe target problems, the experimenter intervened to correct it.

Pegs were separated by 9° of visual angle; the first and the thirdpegs were 4.2° of visual angle from the left and the right edges ofthe display, respectively. The centers of two adjacent disks resid-

ing on a peg were 1.7° of visual angle from each other. Participantswere seated at a viewing distance of 82 cm from the computerdisplay.

Apparatus. Eye movements were monitored by an SR Re-search EyeLink II eye-tracker sampling at 500 Hz. Viewing wasbinocular, but only the right eye was monitored. Stimulus presen-tation and response collection were controlled by SR ResearchExperiment Builder software. The eye-tracker and display monitorwere interfaced with a 3-GHz Pentium 4 PC, which controlled theexperiment and logged the position of the eye throughout all trials.

Results and Discussion

One participant was excluded from the data for Move 4, and adifferent participant was excluded from the data for Move 16, inboth cases because their latency was more than two standarddeviations from the mean for that move in the critical problem (forMove 4, M � 2.1 s, SD � 1.9 s, outlier � 10.3 s; for Move 16,M � 3.4 s, SD � 4.8 s, outlier � 24.2 s). The outcomes of theinferential tests we describe below did not change if these partic-ipants were included.

Figure 6 shows response latencies for the first 32 of the 63moves of the target problems. The solid line represents the criticalproblem and the dashed line represents the average of the twocontrol problems. The pattern in Figure 6, with latency spikes ofvarying heights on Moves 1, 5, 9, . . . is consistent with resultsfrom previous studies of the Tower of Hanoi (e.g., Anderson et al.,1993). The latency spikes reflect application of the goal-recursivealgorithm, with the height of a spike proportional to the number ofdisks in the tower to which the algorithm is applied. Figure 1bshows an example of eye movements on one such move. Thispattern, in which the eyes move from one disk of a tower to its goaldestination, then to the next disk up and its goal destination, and soon, offers converging evidence for use of goal recursion on suchmoves. We also note that, on remaining moves (those other than 1,5, 9, . . .), which are fast enough to suggest they require relativelylittle “thinking,” participants first fixated the source peg for themove and then the destination peg for the move in 76% of cases.This high proportion is consistent with findings suggesting that eye

Figure 6. Response latencies for the first 32 moves of target problems in Experiment 1, separated by move andproblem type (critical, control). The different models predict latency increases on different moves in the criticalproblem compared with the control problems. The ordered-goals model predicts increases on Moves 12, 16, andpossibly 4; the primed-goals model predicts an increase on Move 4; and the selective-attention model predictsan increase on Move 12.


movements in everyday tasks, such as making coffee or preparinga sandwich, are closely associated with the steps of routinizedperformance as it unfolds (Land & Hayhoe, 2001).

For each of the diagnostic moves we discussed above—Moves4, 12, and 16—we compared latency on the critical problem withaverage latency for the two control problems. On Move 4, latencyin the critical problem (M � 1.70 s, SD � 0.54 s) was not reliablyfaster than latency in the control problems (M � 1.60 s, SD �0.53 s), t(21) � 0.88, p � .39. On Move 16, latency in the criticalproblem (M � 2.38 s, SD � 1.29 s) also was not reliably fasterthan latency in the control problems (M � 2.19 s, SD � 0.86 s),t(21) � 0.54, p � .60. On Move 12, latency in the critical problem(M � 5.23 s, SD � 5.77 s) was substantially slower than latencyin the control problems (M � 1.16 s, SD � 0.37 s), t(22) � 3.3,p � .01. We also note that among nondiagnostic moves, Move 1was faster in the critical problem because that problem started withone fewer disk, which reduces the number of recursions of thegoal-recursive algorithm, and on Move 17, the trend was forlatency to be faster in the critical problem, but the effect was notreliable.

The pattern of results on diagnostic moves supports theselective-attention model, which predicted a latency differenceonly on Move 12. The pattern conflicts with the disk-to-peg formof the ordered goals model and with the primed-goals model,which both predicted a latency difference on Move 4. The patternalso conflicts with the peg-to-peg form of the ordered-goals model,which predicted a difference on Move 16—under an assumptionwe noted earlier, that on this move the system is guided by thesame goals it formulated before we added Disk 3a to the problem.If, instead, the system reanalyzed the problem, by applying thegoal-recursive algorithm to all six disks, the new plan wouldspecify the correct goal for Move 16 (AB, as in the controlproblem) and thus predict no latency increase. Reanalysis did notseem to occur on Moves 4–11 or 13–15, where latencies did notdiffer from the control problem. It probably also did not occur onMove 12, where latency was still somewhat faster than Move 1latency on control problems (see Figure 6), which estimates thetime needed to recurse over six disks. Nonetheless, to the extentthat participants could have reanalyzed the problem, the caseagainst the peg-to-peg model is weakened, and we develop astronger test in Experiment 3.

The moves that participants actually made are consistent withthe latency data. On Move 4, the disk-to-peg ordered-goals modeland the primed-goals model both predict that the current goalshould have been 3C. In the critical problem, this goal should havetriggered goal-recursion to compute how to achieve 3C, whichshould have led to 1CA as the actual move (the relevant state is theone in Figure 3b after 1CB is made). Instead, all participants madethe move predicted by the selective-attention model, which is3aAC. Eye movements also suggest that there was no intention toachieve 3C. Had participants intended to move Disk 3, they shouldhave fixated on Disk 3, which was the second disk from the top atthat point in the critical problem. Instead, participants typicallyfocused on the top disk (3a). By two-tailed sign test, there was noeffect of problem type (critical, first control) on the frequency offixations on the top-most disk of the tower at that point in theproblem ( p � .22). Participants were not less likely to fixate thetop disk (3a) in the critical problem than the top disk (3) in the firstcontrol problem.

Associated with the latency increase on Move 12 was a high rateof illegal move attempts in which the participant picked up Disk 3and tried to place it on Peg B, even though Peg B already held Disk3a, which was smaller (the relevant state is the one in Figure 3dafter 1BA is made). Thirteen of 22 participants (59%) attemptedthis move. In each case, the experimenter intervened once theparticipant had released Disk 3 on Peg B, after which the partic-ipant returned Disk 3 to Peg C, and then, of his or her own accord,made the correct move, which was 3aBC. (The time taken up bythe illegal move attempt and the resulting intervention were notincluded in the latency data for Move 12. When an illegal movewas attempted, we started timing the subsequent legal move fromrelease of Disk 3 on Peg C, which was the action that undid theillegal move attempt.) Eye movements suggest that four additionalparticipants (18%) intended to perform the illegal move, becausethey fixated on Disk 3 before fixating on and moving Disk 3a. (Aswe noted earlier, participants usually fixate a disk before theymove it. On Move 12 in the control problems, for example,participants fixated the to-be-moved disk in 91% of cases.) Thishigh proportion of illegal move attempts and intentions (77%combined) suggests that disk size was not a particularly salientfeature of participants’ representations of the problem—eventhough disk size was a relevant task constraint—and that otherinformation was largely guiding move selection.

We also examined latency for illegal move attempts in thecritical problem, timed from release of Disk 1 on Peg A to releaseof Disk 3a on Peg B, to assess whether participants may havehesitated before trying to make this move. Latency for the illegalmove attempt (M � 1.59 s, SD � 0.92 s) was not reliably higherthan latency on Move 12 in the control problems (M � 1.36 s,SD � 0.4), t(12) � 1.22, p � .25. Thus, there was no evidence tosuggest that participants detected a problem with the move beforethey tried to make it.

The eye-movement data also support the selective-attentionmodel and not plan-based models. As we described earlier, theselective-attention model predicts that the eyes should move fromthe just-displaced disk to the peg holding the new tower, whereasplan-based models predict that the eyes should move from thejust-displaced disk to the peg associated with the next goal toretrieve—and in some states, these pegs are different (seeFigure 5). Out of 198 observations (three for each of three prob-lems for each of 22 participants), 91% were consistent with theselective-attention model, and 9% were consistent with a goalguiding the next step of processing. This evidence is especiallyproblematic for the primed-goals model, which makes the strongprediction that the relevant disk has to be attended to cue retrievalof the associated goal. This evidence is also problematic for theorder-goals model under the assumption that the system fixates thedisk or peg it is thinking about at the moment.

To evaluate the extent to which participants were explicitlyaware of any discrepancies between the puzzle display and theirrepresentation of it after the changes in the critical problem, theexperimenter asked them a structured series of questions at the endof the session (5–7 min after the changes occurred). First, theparticipant was asked a general question as to whether he or shenoticed anything unusual. Second, the participant was told thatsomething had changed on the display during one of the problemsand was asked if he or she had noticed. Finally, the participant wastold what had changed on the display and when and was asked if


he or she had noticed. The answers indicate that no participantswere aware of the display changes in the critical problem and thatnone noticed the change in number of disks during the criticalproblem, from five to six. Moreover, the 13 participants who triedto make an illegal move on Move 12 viewed this error as their ownmistake and did not attribute it to any external changes.

One specific behavioral episode also bears on the question ofparticipants’ awareness of our display changes. On Move 4 of thecritical problem, one participant reported to the experimenter thathe had made an error and asked if he could undo his steps. (Thesoftware did not allow undoing of multiple steps, so the problemwas terminated soon after and the participant’s data excluded.) Hehad not made an error, but he had reanalyzed the problem usinggoal recursion, as indicated by his eye movements, which wouldhave suggested that he had made an error—on Move 1, by movingDisk 1 to the wrong peg for what was now a six-disk problem. Hehad also reanalyzed the problem state on Move 4 in the precedingcontrol problem, suggesting that reanalysis was a strategic choiceof some kind, and that reanalysis drove the “error” detection, notthe other way around. Thus, in this one case the effect of thedisplay change was in some sense detected, but not perceptually,and then not correctly attributed.

In sum, performance on the critical problem was basically thesame as on the control problem—except for Move 12, on whichlatencies were several times longer in the critical problem (5.2 s,compared with 1.2 s; see Figure 6). This effect on Move 12 waspredicted by the selective-attention model but also by the peg-to-peg form of the ordered-goals model. The disk-to-peg form of theorder-goals model and the primed-goals model predicted effects onMove 4, for which there was no behavioral evidence. The peg-to-peg form of the ordered-goals model predicted an effect on Move16, on the assumption that participants did not reanalyze the fullproblem state after we added the new disk. Finally, the highproportion of illegal move attempts (59%) on Move 12, after weexchanged locations of two disks, suggests that disk size was notsaliently coded in participants’ internal representations.

Experiment 2: Deletion of Two Disks

Experiment 1 suggested that plans stored in memory were notguiding behavior, but it also differed from an important prior studyin a way that could limit its generality. In the materials used byAnderson and Douglass (2001), disks and pegs were labeled on theproblem display with letters and numbers, in addition to varying interms of visuospatial characteristics (size and location) as they didhere. These verbal labels represented an alternative source ofinformation that participants could have made use of to code goalsin memory. If use of plans is under strategic control, this avail-ability of this extra information could have made a plan-basedstrategy more attractive. Thus, in Experiment 2, we labeled thedisks and pegs.

We also altered the manipulation in the critical problem to focuson another hypothesis that seems to arise from plan-based models,which is that the system will detect and react to a discrepancybetween the number of internal object representations and thenumber of external object referents. Such a discrepancy occurredin the critical problem we used in Experiment 1, but here wesought to enhance it. Instead of adding one disk, we deleted twodisks, turning a seven-disk problem into a five-disk problem. This

should lead to intermediate states in which, according to plan-based models, there are more goals represented internally thanthere are disks to which they are relevant; the maximum ratio ofgoals to disks is 3 to 1, which would seem to be a salientdiscrepancy. The hypothesis that the system will detect such dis-crepancies may be less direct a consequence of plan-based repre-sentations than predictions linking goals to specific moves, whichwe tested in Experiment 1 and test again here, but it nonethelessrepresents a converging operation. If performance reflects neitherthe information in specific goals nor discrepancies between thenumber of goals and the number of objects to which they refer, thiswould strengthen the case that goals are simply not represented.

Finally, the saccade-contingent changes we make here are moredramatic than they were in Experiment 1. Here we changed theheight of the initial tower by two disks instead of one within thefirst few moves and changed the locations of several disks con-currently with each disk deletion. These more extensive changesshould help test the limits of our procedure in terms of whatsaccade-contingent changes will go unnoticed by participants. Ingeneral, the more that one can manipulate in a problem displaywithout leading to immediate awareness of the manipulation, thegreater the power of the procedure to create discrepancies betweeninternal and external representations useful for elucidating whatthose internal representations are.

We used letters to label the disks and numbers to label the pegs,as shown in Figure 7. We used letters for disks instead of numbersto discourage participants from thinking about disks in numericterms, hoping to prevent participants from discovering a heuristic,based on the parity of the number of disks in a tower, that cansubstitute for the goal-recursive algorithm.

There were two sets of saccade-contingent display changes inthe critical problem. First, between Moves 2 and 3, we shortenedthe initial tower by one disk. To do this without introducing a gapin the size ordering of the remaining disks, we deleted the smallestdisk on the display, then moved the second smallest disk into itsold position, then moved the third smallest disk into the oldposition of the second. Figure 7 shows the problem state beforeand after these changes. In Figure 7b, which shows the state afterthe change, Disk P has been deleted from Peg 3, Disk K hasreplaced it, and Disk E has replaced Disk K on Peg 2. Thesechanges occurred as the participant made the saccade from Disk Kto Disk P to pick up Disk P and place it on Disk K. The second setof changes, which used the same method to shorten the initialtower by a second disk, occurred between Moves 4 and 5. Figure 7shows the problem state before and after these changes (seeFigures 7c and 7d, respectively), which were triggered when theparticipant made the saccade from Disk T to Peg 2 to computewhere to move the two-disk tower located there. This second diskdeletion served to preserve the original final state of the problem,as the exchange of two disk locations did in Experiment 1.

Diagnostic Moves

The disk-to-peg form of the ordered-goals model and theprimed-goals model predict a latency increase on Move 4 in thecritical problem; the peg-to-peg form of the ordered-goals modelpredicts latency increases on Moves 9 and 16 in the criticalproblem; and the selective-attention model predicts no latency


increases in the critical problem. In the following, we considerthese moves and model predictions in detail.

Move 4. We illustrate Move 4 in the critical problem in termsof Figure 7. The problem state shown in Figure 7b is the oneimmediately preceding Move 3, which transfers Disk K from Peg3 to Peg 2, creating a two-disk tower on Peg 2 and leaving Peg 3empty. Move 4 then transfers Disk T from Peg 1 to Peg 3.

The predictions for Move 4 correspond to those in Experiment1, although here the manipulation should be somewhat strongerbecause there are additional display changes. As in Experiment 1,the ordered-goals model predicts a latency increase on Move 4under the assumption that a goal codes the to-be-moved disk andits destination peg. Under this assumption, the current goal at thispoint is E3 (i.e., to move Disk E to Peg 3). After the displaychanges in the critical problem, this goal cannot be directlyachieved, because Disk E is now covered by Disk K, so latency onthis move should reflect time for the system to use goal recursionto compute how to achieve the current goal. Moreover, unlikein Experiment 1, Disk E is now on a different peg than it was whenthe goal was encoded (Peg 2 instead of Peg 1). Thus, if the goalrepresentation includes any information about the original locationof the to-be-moved disk, such as a peg label or spatial information,this would also be in conflict with the external problem state,contributing separately to an increase in latency, if only by trig-gering visual search to find Disk E.

The primed-goals model also predicts a latency increase onMove 4, because any goal primed for retrieval at this point shouldnot be directly achievable. The retrieval cue should either be DiskE, which was most recently uncovered but is now to be found ona different peg, or Disk T, which is the top disk on the relevant

tower. The goal for Disk E cannot be directly achieved becauseDisk E is covered by Disk K. The goal for Disk T, which shouldbe T2, cannot be directly achieved because Peg 2 holds smallerdisks. Whichever goal is retrieved, then, should trigger goal recur-sion that adds to response latency.

The selective-attention model predicts no latency increase onMove 4, because there is no conflict between the external andinternal problem representations. There is now a new two-disktower on Peg 2, which was created by Move 3, and recognition ofthis new tower should lead the system to displace another diskfrom the old tower, which is on Peg 1, to the empty peg.

Moves 9 and 16. The problem states immediately prior toMoves 9 and 16 are shown in Figure 8. Move 9, shown inFigure 8a, transfers Disk E from Peg 3 to Peg 2. Move 16, shownin Figure 8b, transfers Disk I from Peg 1 to Peg 3.

The peg-to-peg form of the ordered-goals model predicts alatency increase on both moves, under the assumption that partic-ipants are able to detect substantial discrepancies between thenumber of goals represented internally and the number of relevantobjects in the environment. On both moves only one disk remainsof the initial tower (Disk I), and if the internal representation werealigned with this state then only one goal should remain (Goal 13).Instead, these states occur at points in the critical problem at whichthere should be three goals referring to the initial tower. This 3:1ratio of goals to disks was higher than the maximum 2:1 ratio ofgoals to disks that occurred in Experiment 1, and we supposed thatit would offer a reasonable test of whether this kind of represen-tation discrepancy would be detected. In case detection interactedfor some reason with whether or not the current goal is directlyachievable, the two moves offer a useful distinction, in that Goal

Figure 7. Saccade-contingent display changes in the critical problem in Experiment 2. In this instance of thecritical problem, the initial tower was on Peg A, and the instruction was to move the tower to Peg C. a: Theproblem state after Move 2, which is K12 (transfer Disk K from Peg 1 to Peg 2). The display changes that followare to delete Disk P and move Disks E and K. b: The problem state immediately after these changes. c: Theproblem state after Move 4, which is T13. The display changes that follow are to delete Disk K and move DisksE, T, and D. d: The problem state immediately after these changes.


13 is achievable on Move 16 (this is the move shown in Figure 8b)but not on Move 9.

Method

Participants. Thirty-five undergraduate students from theMichigan State University psychology subject pool participated inexchange for course credit. None had participated in Experiment 1.All had normal or corrected-to-normal vision. A participant wasincluded in the analysis if he or she made no errors on diagnosticmoves in the control problems and no more than two errors on thecombined set of 26 nondiagnostic moves from the first 16 movesof the control problems (these 16 are the moves we examine in theResults and Discussion section). Twenty participants met thiscriterion.

Apparatus and procedure. These were the same as in Ex-periment 1 except that (a) at the start of the session, the experi-menter referred to the disks and pegs by their verbal labels inaddition to using referential gestures, in describing the goal-recursive algorithm; and (b) the saccade-contingent display changeinvolved transforming the critical problem from a seven-disk puz-zle into a five-disk puzzle, as described previously.


Two participants were excluded from the data for Move 4, inboth cases because their latency was more than two standarddeviations from the mean for that move in the critical problem(M � 3.0 s, SD � 4.4 s, outliers � 18.8 s and 12.0 s). Theoutcomes of the inferential tests we describe below did not changeif these participants were included.

Figure 9 shows response latencies for the remaining participantsfor the first 16 moves of the target problems. The solid linerepresents the critical problem and the dashed line represents theaverage of the two control problems. On the three diagnosticmoves, latencies were not reliably affected by problem type (crit-ical, control), t(17) � 1.37, p � .19 for Move 4, t(19) � 1 forMove 9, and t(19) � 1.43, p � .17 for Move 16; on all threemoves, the trend was for latency to be faster in the critical problemthan in the control problem.

The moves that participants actually performed are consistentwith the latency data. On Move 4, the disk-to-peg ordered-goalsmodel predicts that the current goal should have been E3. In thecritical problem, this goal should have triggered goal recursion that

ultimately led to the next move being K21 (the relevant state is theone in Figure 7b after K32 is made). Instead, all participants madethe correct next move, which was T13. The primed-goals modelalso predicts K21, based on either of the two goals that could beprimed for retrieval at this point, which are E3 and T2.

To evaluate the extent to which participants were aware ofrepresentational discrepancies stemming from the saccade-contingent display changes, which were more substantial here thanin Experiment 1, we asked the same debriefing questions we did inExperiment 1. The answers of nine of the 20 participants (45%)indicated that they had not noticed the display changes. Seven ofthe 20 participants (35%) reported, in response to the first ques-tion, about whether they had noticed anything unusual, that theoverall height of a tower had become shorter. Four of the 20participants (20%) reported that disks had been deleted; two ofthese four reported this in response to the first question and theother two in response to the third question, when they had beentold what had changed and when.

The two-disk change in height of the initial tower was quitesalient perceptually, at least to an observer who knew it happened,

Figure 8. Two states in the critical problem of Experiment 2, with the correct move in each state (indicated byarrows). a: The state before Move 9, which is E32 (transfer Disk E from Peg 3 to Peg 2). b: The state beforeMove 16, which is I13.

Figure 9. Response latencies for the first 16 moves of the target problemsin Experiment 2, separated by move and problem type (critical, control).The ordered-goals model predicts increases in the critical problem com-pared to the control problems on Moves 9, 16, and possibly 4; theprimed-goals model predicts an increase on Move 4; and the selective-attention model predicts no increases.


so it may not be surprising that some participants were able toidentify it. However, the change in height was not the only cluethat something had changed about the problem; another was thatthe critical problem took much less time to solve than it shouldhave as the seven-disk problem it started as—31 moves, comparedwith 127 moves. To the extent that participants had expectationsfor problem duration based on the number of disks in the initialstate, these would have been violated. Indeed, two participantsmade time-related comments during the debriefing. One of thenine who did not seem to notice any changes reported that eitherthe problems were getting easier or she was getting better, and oneof the five who reported noticing the disk deletions noted that aproblem did not take as long as she thought it should have. In thelatter case in particular, a violation of temporal expectancy couldthen have supported an inference that something had changed onthe problem display.

None of the 20 participants gave any indication of havingnoticed that the disks had moved around during the displaychanges, even though both disk size and disk labels were availableas cues to this change. This extends the finding from Experiment1 that participants did not seem to code disk size in their repre-sentations of the problem state (as reflected in the large number ofillegal move attempts on Move 12). It may be that in general,features that discriminate among disks are not paid much attention,an issue to which we return in Experiment 3.

In sum, performance on the critical and control problems wasbasically the same, on all moves (see Figure 9). As in Experiment1, this conflicts with predictions based on what the current goalshould have been, had goals been directing behavior. The disk-to-peg form of the ordered-goals model and the primed-goals modelpredicted a latency increase and an error on Move 4. There was noevidence for a latency increase, except for two outliers, and therewere no errors. In this experiment we also asked whether adiscrepancy between the number of internal goals and the numberof external referents would be detected. The discrepancy wouldhave been greatest on Moves 9 and 12 of the critical problem, butthere was no latency increase relative to the control problem ineither case. Finally, we saw no evidence that verbal labels for disksand pegs facilitated use of a plan-based strategy, or even that theselabels were encoded in participants’ representations.

Experiment 3: Making a Move on theParticipant’s Behalf

In Experiment 3 we contrast the selective-attention model withthe peg-to-peg form of the ordered-goals model, in which weassume that a goal codes the source and destination pegs for amove, but no information about the to-be-moved disk. This modelcould accommodate the results of Experiment 1 under the assump-tion that people formulated new goals after the display change, andthe results of Experiment 2 under the assumption that the systemdoes not detect discrepancies between the number of goals repre-sented internally and the number of disks on the puzzle display.Here we implement a different kind of display change, in which wemake a move on the participant’s behalf, and ask what move theparticipant makes next.

We also further examine what details about the problem stateare coded in participants’ problem representations. Experiments 1and 2 suggested that neither disk size nor disk labels were coded.

Here, we ask about the height of a disk on a peg. The move wemake on the participant’s behalf empties the location in space thatthat disk had been occupying. If the height of that location is notcoded, we would expect the next lower disk on that peg to be easilyaccepted as a “replacement” for the disk that we moved, with littleevidence that participants notice the difference. This would sup-port our assumption that attention can be allocated to a towerwithout being allocated specifically to its constituent disks.

The move we make for participants is illustrated in Figure 10.(The critical problem in this experiment involved six disks; in therepresentative problem that is the basis for Figure 10, the initialtower would have been on Peg A and the final tower on Peg C.)The software moves Disk 3 from Peg C, as located in Figure 10a,to Peg A, as located in Figure 10d. This is Move 20 of the problemand occurs in two stages, as the participant makes Move 19.Figure 10a shows the problem state before Move 19, which is1AB. As the participant saccades to Peg A to pick up Disk 1, thesoftware deletes Disk 3 from Peg C, as shown in Figure 10b. Theparticipant then places Disk 1 on Peg B, creating the state in Figure10c. As the participant then saccades away from Peg B, Disk 3appears on Peg A, as shown in Figure 10d. We moved Disk 3 intwo stages like this to avoid changing the target of the saccade thattriggers the change, so as to reduce the likelihood that participantswould detect the change (Henderson & Hollingworth, 2003).

Diagnostic Move

There is one diagnostic move in this experiment, which is Move21, immediately following the display change. In context of Figure10d, the correct move at this point—the one that keeps the systemon the optimal solution path—is 1BC.

The main diagnostic measure here is not latency but the moveactually made. For the peg-to-peg model, the goal is CA, to movethe top disk from Peg C to Peg A. Peg C holds Disk 4, and Peg Aholds Disk 3 (see Figure 10d), so this move is illegal, but partic-ipants may try it anyway, if they did not code the height of Disk3 on Peg C before we moved it and now try to move the diskunderneath it. Following any such illegal move attempts, the moveactually made will depend on whether the system reanalyzes theproblem to formulate a new plan. If so, we would expect thecorrect move, 1BC. If not, such that the model retains Goal CAfrom before the display change, the next move should be 1BA, thefirst step of a sequence that clears Disk 3 from Peg A (1BA, 2BC,1AC, 3AB, 1CA, 2CB, and 1AB). However, because 1BA undoes1AB, which was the participant’s last move, it might be discour-aged by the “don’t-undo” heuristic.

The selective-attention model predicts, qualitatively, that thesystem will recognize the two-disk tower on Peg B (see Figure10d) and then want to perform the displace step of the recognize-displace-compute routine. The tower currently being disassembledis on Peg C, so the system may try to displace Disk 4 from Peg Cto Peg A. Thus, as under the peg-to-peg model, participants maytry to make the illegal Move 4CA, assuming, again, that they didnot code the height of Disk 3 on Peg C. Following any such illegalmove attempts, the move actually made will again depend onwhether the system reanalyzes the problem. If so, we would expectthe correct Move 1BC. If not, the system might first perform amore limited reanalysis to try to bind the displace operation to adisk that can be legally moved. Disk 3 is the only candidate, as


Disk 1 never belongs to an old tower, so this limited reanalysiswould predict instances of the error Move 3AC. There would be noobvious explanation of 3AC in terms of the peg-to-peg model.

Method

Participants. Thirty undergraduate students from the Michi-gan State University psychology subject pool participated in ex-change for course credit. None had participated in the previousexperiments. All had normal or corrected-to-normal vision. Aparticipant was included in the analysis if he or she made no errorson Moves 17–24 in the control problems. Twenty-two participantsmet this criterion.

Apparatus and procedure. These were the same as in Ex-periment 1 except that (a) the critical problem was a six-diskpuzzle, and (b) the saccade-contingent display change involvedmaking a move on the participant’s behalf, as described above.


The state after the display change in the critical problem isshown in Figure 10d. From this state, 10 of 22 participants (45%)tried to make the illegal Move 4CA. Latency for the attempt, timedthrough release of Disk 4 on Peg A (after which the experimenterintervened) was about as long (M � 2.0 s, SD � 1.5 s) as latencyfor Move 21 in the control problem for these participants (M �1.8 s, SD � 0.9 s), t(9) � 0.38, p � .72. Thus, there was nohesitation to try to move Disk 4 in place of Disk 3. In these cases,then, participants seem not to have coded the height, or any otherdetails, of Disk 3. These illegal move attempts do not distinguish

between models, in that the system could have been trying eitherto achieve Goal CA or to displace Disk 4.

In terms of the move actually made (after any illegal moveattempts), no participant made the error Move 1BA, but 13 of 22participants (59%) made the error Move 3AC (six of these afterfirst trying the illegal move). Move 3AC was predicted by ouranalysis above suggesting that, once the system found it could notdisplace Disk 4, it would bind the displace operation to Disk 3, theonly other candidate for a “displaceable” disk at this point. Thus,these moves seem to have a unique interpretation in terms of theselective-attention model.

The moves made after the error Move 3AC are also illuminat-ing. In four of the 13 cases, the participant undid the error and thenmade the correct Move 1BC—but in the remaining nine, theparticipant continued by transferring the two-disk tower on Peg B(see Figure 10d) to Disk 3 on Peg C. This sequence is consistentwith the selective-attention model if Move 3AC had been theeffect of a displace operation, as we suggested previously. That is,we would say that these nine participants recognized the newtwo-disk tower on Peg B, found a disk to displace (Disk 3 insteadof Disk 4), then moved the new tower to the just-displaced disk, allin accord with the recognize-displace-compute routine. The effectof this sequence was to recreate a problem state from earlier in thetrial, before the display change. From this recreated state, all nineparticipants made the same moves they had earlier, triggering asecond instance of the display change. Six then made the correctMove 1BC, but three made the loop a second time. At a generallevel, apart from the specifics of the selective-attention model, thislooping behavior seems consistent with a situated view in that

Figure 10. Saccade-contingent display changes in the critical problem in Experiment 3. The display changeshave the effect of making Move 20, which is 3CA (transfer Disk 3 from Peg C to Peg A), in two steps. a: Theproblem state before Move 19, which is 1AB. As the participant makes the saccade to Peg A to pick up Disk1, Disk 3 is deleted from Peg C. b: The problem state immediately after this change. c: The problem state afterMove 19. The participant has just released Disk 1 on Peg B, and gaze is still directed at Peg B. The next saccadecauses Disk 3 to reappear on Peg A. d: The problem state immediately after this change.


performance was strongly guided by the environmental changeswe introduced and apparently not so strongly by memory formoves or states of even a moment ago.

All participants ultimately made the correct Move 1BC, but onlyfive of 22 made it directly, without intervening illegal-move at-tempts (4CA) or errors (3AC). Latency in these five cases wassubstantially greater (M � 10.2 s, SD � 4.6 s) than it was for Move21 in the control problem for these participants (M � 1.5 s, SD �0.9 s), t(4) � 4.5, p � .01. This latency increase may havereflected reanalysis of the problem after the display change, whichwas necessary to identify the correct move. At almost 9 s, theincrease may have been large enough to reflect mental simulationof the missteps that other participants made physically, in whichcase the trigger for reanalysis in all cases may have been this kindof search through the problem space.

In sum, after the display change, many participants (45%) triedto make an illegal move, indicating they had not coded the heightof the disk we moved on their behalf and simply tried to move thedisk underneath. These illegal move attempts could have beenguided by a peg-to-peg goal or by an operation to displace a diskfrom an old tower. For the first legal move after the displaychange, most participants (59%) made an error move that weinterpret in terms of the system finding a way to bind the displaceoperation after the original binding failed.

General Discussion

Traditional research on routine behavior has often appealed tocomplex internal representations such as plans (e.g., Ernst &Newell, 1969; Miller, Galanter, & Pribram, 1960) and has testedsuch accounts using tasks for which plans seem ideally suited:those with complex rules but simple, static environments. TheTower of Hanoi is a canonical example, and current models ofperformance on this task assume it is guided by goal structures(e.g., Altmann & Trafton, 2002; Anderson & Douglass, 2001). Ourresults indicate that the orderly nature of routine performance onthis task results not from plans, and suggest instead that processingis guided by a structured environment triggering the operation ofsimple, abstract rules whose parameters are bound through per-ception and attention.

In the following, we first summarize our evidence on plan-basedmodels and revisit some previous evidence offered in their favor totry to reconcile the conflict. We then summarize our evidence onthe selective-attention model. Finally, we explore implications ofour results for everyday task performance and for neuropsycho-logical deficits traditionally interpreted in terms of goal-directedfunctioning.

Evidence Concerning Plan-Based Models

The evidence from our three experiments suggests that there islittle if any role for move-specific goals in governing performancein this task environment. Perhaps the most direct evidence con-cerning move-specific goals comes from the situations we createdin which what should have been the current goal predicted behav-ior that we did not observe. In Experiments 1 and 2, we createdsituations in which the current goal would have been achievable inthe control problems but not in the critical problem. This in turnshould have led to higher response latency on that move in the

critical problem, as the system used goal recursion to compute howto achieve the goal. With one exception, we saw no such increases.The one exception was Move 12 of Experiment 1 (see Figure 6),but here the latency increase was also anticipated under theselective-attention model. Moreover, in all these situations, as wellas in Experiment 3, move-specific goals predicted moves thatparticipants did not actually make. An important caveat on thisanalysis is that to test the ordered-goals model in particular, we hadto make assumptions about what information is coded in goals,going beyond the level of detail at which the model was previouslyspecified (Anderson & Douglass, 2001). Although we tested twosets of assumptions, leading to the disk-to-peg and peg-to-pegforms of the model, there may be others that maintain the basicfunctionality of goals but also accommodate the present empiricalresults.

A second line of evidence is an apparent lack of specificdisk-related information coded in participants’ representation ofthe problem state. In Experiment 1, on a move after a displaychange, most participants tried to move a wrong-sized disk ille-gally without hesitation, continuing the move to the point wherethey dropped a larger disk on a smaller disk. In Experiment 2, wemoved several disks around during a saccade at two points in thecritical problem, with no effect on response latencies in either case.Here the disks were also labeled, suggesting that participants didnot code label information either. In Experiment 3, many partici-pants again tried to make an illegal move, even though the diskthey selected to move was both larger and in a different locationthan the disk they would have selected had we not moved it forthem. Given that neither the size of a disk, nor any labels, nor thelocation of a disk seem to be salient enough to prevent a disk frombeing inappropriately selected for moving, it is unclear how thesystem could make any use of goals that hypothetically codedisk-specific information. This argument applies to goals thatcode disks and thus does not apply to the peg-to-peg form of theordered-goals model.

A third line of evidence is the absence of any effect of discrep-ancies between the number of goals that the system should havebeen maintaining, in some sense, and the number of objects in thetask environment to which those goals were relevant. In Experi-ment 2, we created situations in which the ratio should have been3:1, yet there was no effect on response latency. This evidence isindirect in the sense that it rests on assumptions about the natureand operation of discrepancy-detection mechanisms. In Experi-ment 3, however, we learned that participants were ultimately ableto detect when a move (3CA) led them back to an old problemstate and make the correct move instead. Thus, as part of a largerpattern, the absence of evidence that participants detected anexcess of goals seems easiest to explain in terms of an absence ofgoals.

Other studies have offered evidence for goals in the Tower ofHanoi, which we try here to reconcile with our results. Andersonand Douglass (2001) reported higher response latency at a point inthe problem where a goal hypothetically had to be retrieved,compared with a point at which no goal had to be retrieved. In theirprocedure, participants performed two types of actions: moves, asin our procedure, and goal-posting actions, with which participantsregistered a goal with the experiment software as they formulatedit. Anderson and Douglass compared latencies for two differentgoal-posting actions. In the first case, a goal referring to Disk 3


was posted immediately after a goal referring to Disk 4 had beenposted (their “Under 4” category). In the second case, a goalreferring to Disk 3 was posted immediately after a goal for Disk 5hypothetically had been retrieved from memory (the retrievalsubcategory of their “Under 5” category); this goal would not havebeen achievable at this point and therefore would have triggeredthe goal-recursive algorithm. Response latency was higher in thesecond case, which Anderson and Douglass took as evidence oftime to retrieve the goal for Disk 5. However, this effect couldhave been due to other operations required to set up the goal-recursive algorithm. In our model, these would have includedattending to the newly recognized tower after displacing a disk,and finding its largest disk. Such operations would have beenincluded in the second case, where timing started on the previousmove (the displace step), but not in the first case, where timingstarted with the previous goal posted during application of thegoal-recursive algorithm.

Evidence Concerning the Selective-Attention Model

We characterize performance in this task environment in termsof an abstract control routine with parameters bound by perceptualand attentional operations. In each of our experiments, for a set ofdiagnostic moves, we identified how the parameters of the routinewould be bound—which new tower had been recognized, whichold tower was being disassembled, and which disk was or wouldbe displaced—and how execution of the routine accounted for thedata. Two moves are of particular interest. For Move 12 of Ex-periment 1, we created a situation in which the top disk of the oldtower could not legally be displaced after the display change. Weobserved increased latency and a large proportion of illegal moveattempts, consistent with difficulties executing the displace oper-ation. For Move 21 of Experiment 3, we created a similar situationby making Move 20 on the participant’s behalf, which again led toa large proportion of illegal move attempts. The illegal moveattempts in both situations could have been guided by peg-to-peggoals as well as displace operations. However, in Experiment 3,the move that participants actually made after the display change(and after any illegal move attempts) was most often an error thatsuggested the system had found a way to bind the parameters ofthe displace operation. This move was often followed by thethree-move sequence that the recognize-displace-compute routinewould predict had that move been the effect of a displace opera-tion.

With respect to the level of detail coded in people’s represen-tations of objects in the environment, we found little evidence thatthese include features that discriminate among disks within atower. In Experiment 3, for example, participants often took Disk4 on the lowest level of a peg as a substitute for Disk 3 one levelup, which was the disk we had moved on their behalf. Thissuggests that disk size and disk height were not coded in thoseparticipants’ representations of the problem state. Similarly, inExperiment 2, the sizes and labels of several disks were affected bythe display changes, but this seemed not to affect the performanceof any participants. These observations are consistent with the ideathat people are able to link an abstract task-relevant concept (e.g.,the just-displaced disk) to the associated object in the environmentthrough an index or pointer, rather than coding the object itself(Pylyshyn, 2000).

Implications for Everyday Tasks

The Tower of Hanoi is only one task environment, but the ideaof planless action execution might apply more broadly—for ex-ample, to everyday activities, such as preparing coffee or packinga lunchbox. In such tasks, a given action, instead of being linkedto the previous action in an internal representation, could again betriggered by perception of the effects of the preceding action. Forexample, the act of adding creamer could be triggered by the sightof black coffee in a cup, much as a robot’s reach for a soda cancould be triggered by its proximity to the table on which the can isresting (Brooks, 1991).

The everyday behavior of nonhuman organisms is often inter-preted in such situated terms. For example, the stickleback fish, toreproduce, migrates to shallow water, where it selects territory,builds a nest, fights other males, and mates with a female. Thiscomplex behavior might appear to be purposeful or intentional, asif the fish is carrying out one or more plans all linked to the goalof reproducing. And yet, each act is triggered by some environ-mental condition: Migration is triggered by a change in length ofthe day, fights are triggered by another male’s presence, nestbuilding can be taken up where it was left off because one stepcues the next. What is noteworthy about this example is that it wasraised by planning theorists (Miller et al., 1960), who wrote, “It isalmost as if the Plan were not in the organism alone, but in the totalconstellation of organism and environment together” (p. 78). Theyused this example to illustrate what they saw as a contrast with theplanfulness of human behavior. The evidence we have presentedsuggests that this contrast can be in the eye of the analyst. Even ina task environment in which planfulness has particularly high facevalidity, plans do not necessarily guide behavior.

Contemporary models of everyday behavior characterize it tosome extent in situated terms. For example, in Cooper and Shal-lice’s (2000) model, environmental constraints are built into thepreconditions of action schemas. For example, the “pick-up”schema is triggered only when there is an object suitable forpicking up and a free hand available to perform the action. To theextent that this and related models (Botvinick & Plaut, 2004; seealso Cooper & Shallice, 2006) do assume rich internal represen-tations, we note that they are constrained mainly by patient per-formance on everyday tasks (e.g., Schwartz, Reed, Montgomery,Palmer, & Mayer, 1991; Humphreys & Forde, 1999), which, as weargue in the following, can be ambiguous as to what the underlyingdeficits are. The saccade-contingent method we used here, to probeinternal representations by manipulating their hypothetical refer-ents in the world, seems to offer a more precise method forconstraining such models with behavioral data.

Tasks like the Tower of Hanoi and making coffee are perhapsmore amenable than others to a perception-driven analysis becauseof their visual-spatial nature, raising the question of whether sim-ilar principles or analysis could be applied to more abstract tasks.Landy and Goldstone (2007) recently demonstrated that perceptualcharacteristics influenced mathematical reasoning, which, muchlike Tower of Hanoi performance, has conventionally been inter-preted in terms of purely abstract or syntactic operations. The basicresult was that experimental manipulation of notational form andof physical spacing and alphabetic similarity of algebraic variablesaffected the order in which algebraic operations were applied. In asimilar vein, we would say that mathematical notations occur in


the physical world and that understanding how they are perceivedand attended is central to understanding how they are processedgenerally. Thus, for example, long division might involve anattentional index binding the concept of “divisor” to the numbersbelow the division sign. Similarly, algebraic manipulations mightinvolve perception of movements of symbols, such that correctupdating of indexes makes the difference between correct stepsand errors. These ideas are speculations but flow from our con-ception of the selective-attention model and should help to framefuture experimental work.

One qualification on our results is that they characterize perfor-mance only after an initial learning stage, which, for a relativelycomplex task like this, might be the place to look for evidence ofproblem solving and thus of goal-based processing. However, itmay be that, even during initial learning, the representations guid-ing performance are abstract in similar ways to the control routinethat we suggest is the product of learning. In particular, duringexploratory learning, which seems to occur in the Tower of Hanoias suggested by protocol evidence (Anzai & Simon, 1979; Van-Lehn, 1991), the system acts on the world and learns by observingthe results, in a sense using perceptual and attentional operations tobind the parameters of a learning process. In terms of an everydaytask, it may be easier to cook from a new recipe, for example, ifmemory for the order of steps is represented externally as a spatialarrangement of ingredients on the table, perhaps with a utensil likea knife used as a pointer for place-keeping (Kirsh, 1995). Thisexternal “scaffolding” may then be absorbed into the controlroutine that is learned, with spatial cues continuing to mediate theassociative relationship between one step and the next. Thus,although our results directly characterize only the routinized per-formance that follows initial learning, the architectural constraintswe tested here seem to be relevant more generally, suggesting thatinitial symbolic learning processes may themselves be situated inimportant ways.

Implications for Neuropsychology

The Tower of Hanoi is widely used in the assessment of frontallobe function. It has been administered to a variety of clinicalpopulations, such as people with Huntington’s disease (e.g., Rob-ins Wahlin, Lundin, & Dear, 2007), schizophrenia (e.g., Rushe etal., 1999), traumatic head injury (e.g., Vakil, Gordon, Birnstok,Aberbuch, & Groswasser, 2001), obsessive–compulsive disorder(e.g., Cavedini, Cisima, Riboldi, D’Annucci, & Bellodi, 2001),autism (e.g., Bolte & Poustka, 2006), aphasia (e.g., Glosser &Goodglass, 1990), cerebellar atrophy (Grafman et al., 1992), am-nesia (e.g., Xu & Corkin, 2001), and attention-deficit/hyperactivitydisorder (e.g., Aman, Roberts, & Pennington, 1998). The task hasalso been used to study aging (e.g., Head, Raz, Gunning-Dixon,Williamson, & Acker, 2002). This inventory includes only studiesusing the Tower of Hanoi, excluding those using the similar Towerof London puzzle (e.g., Shallice, 1982). The procedure for admin-istering the Tower of Hanoi can vary considerably across studies,complicating any comparative analysis, but perhaps the most rel-evant procedural variable for our purposes is the amount of prac-tice each participant received on the task, which bears on whetherperformance is routinized and thus falls within the scope of ourmodel. Across the studies cited above, the number of trials perparticipant ranged from one to 30, with a median of eight, with

eight being the number of trials each of our participants performed.Thus, on the issue of practice, the external validity of our results isnot grossly undermined.

The usual approach in clinical studies is to interpret Tower ofHanoi performance as an index of the ability to perform high-levelcomputations like goal recursion. Goel and Grafman (1995) took adifferent approach, proposing that performance deficits in patientswere linked not to difficulty with such computations but to diffi-culty managing conflicts between different goals stored in mem-ory. On the basis of our results, we suggest that performancedeficits in patients may not be linked to difficulty managing goalconflicts either but, instead, to deficits in elementary processes likeselective attention.

Goel and Grafman (1995) observed that a counterintuitive back-ward move on the solution path predicted patients’ failure to solvethe problem and, on the basis of this observation, proposed thatparticipants have difficulty inhibiting the global goal—the locationof the tower in the final state of a problem—in favor of the localgoal governing the next move. The idea was that the necessaryinhibitory ability is similar to that needed to inhibit the prepotentresponse in the antisaccade task or to inhibit word reading in favorof color naming in the Stroop task. However, we observed similardifficulties at similar moves in our healthy participants, mainly asthey solved the first training problem. A situated way to interpretthese difficulties is as reflecting a transient fixedness in interpret-ing the functions of puzzle objects. Early in an experimentalsession, after participants first experience the task constraints—that only the top disk on a peg can be moved and that larger diskscannot be placed on smaller disks—they might interpret disks in“negative” functional terms, as objects that block the movement ofother disks. Later in a session, once participants have learned thetask, we assume that they interpret the just-displaced disk, forexample, in more “positive” terms, as the destination for thelargest disk of a newly recognized tower. Thus, successful perfor-mance may depend on a shift in how disks are “seen.” A similarshift is needed for the peg holding the final tower of a problem; ittakes a kind of insight to see that this can serve as a temporaryholder of other disks before the largest disk can be moved there.Evidence for this shift can be found in Anzai and Simon’s (1979)protocol data, which contain the passage, “I wonder if I’ve foundsomething new . . . C will be used often before 5 gets there.”Perhaps, then, patients suffer a less transient functional fixednessthat interferes with their ability to see task-relevant objects in theappropriate light.

Goel and Grafman’s (1995) goal-conflict account can also betested using behavioral measures. If patients have difficulty inhib-iting prepotent responses and therefore act impulsively, as Goeland Grafman proposed, then response latencies for patient groupsshould be faster than response latencies for control groups. How-ever, Rushe et al. (1999) found that response latencies for frontallobe patients were not faster than those of controls. They alsoreported that the effect of goal conflict on a particular movedisappeared after just four trials, even in patients, a pattern theyinterpreted as a practice effect. These two pieces of evidencesuggest that the processes involved in resolving a goal conflictdiffer from the inhibitory processes needed in the antisaccade andStroop tasks. It is difficult to compare Rushe et al.’s results directlywith Goel and Grafman’s results because of differences in meth-odology; for example, Rushe et al. used only a composite perfor-


mance score (which included speed, error measures, problem dif-ficulty, etc.), rather than considering speed and accuracyseparately. Nonetheless, it seems that deficits in the ability torestructure how objects are perceived is one way that higher leveldeficits in Tower of Hanoi performance could arise.

Our model suggests other possible low-level factors that maycontribute to higher level deficits. For example, in solving thepuzzle, it helps to have some kind of conceptual representation ofthe idea that towers need to be transferred from peg to peg (Anzai& Simon, 1979), but how is this representation acquired? Onepossibility is that acquisition depends on sustained attention to theset of disks making up a tower as they are transferred from peg topeg. For example, when a new tower is attended, each of itsconstituent disks may share the identity of being a part of thattower. As that tower is being moved, its disks are distributedacross several pegs but may still have the indexes attached to them,consistent with the multiple-object tracking functionality ofFINSTs (Pylyshyn, 2000). When the disks finally come togetheragain as a tower on a destination peg, this would support acqui-sition of a conceptual representation indicating that a specifictower had just been transferred from one peg to another. In termsof a recent theory of skill acquisition (Taatgen, Huss, Dickison, &Anderson, 2008), the tower before the move and the tower after themove would represent the preconditions and postconditions, re-spectively, of a to-be-learned rule. However, if a patient is unableto assign indexes to objects, or to properly track the disks acrosstime and location, then what he or she would perceive is not atower being transferred but individual disks being moved around.Thus, a low-level attentional deficit could, in principle, lead to ahigh-level deficit in terms of which rules were learned.

Another low-level deficit that could contribute to high-leveldeficits could be a difficulty in acquiring specifically eitherexplicit or implicit knowledge and in knowing when to rely onwhich. People seem to acquire implicit and explicit knowledgein parallel, and separating the contributions of each to perfor-mance is not trivial (Cleeremans & Jimenez, 1998). Yet infor-mal analysis suggests that some moves in the Tower of Hanoican probably be made on the basis of implicit knowledge. Thiscould include the “don’t-undo” and “1-follows-2” heuristicsthat we carried over here from previous models (e.g., Altmann& Trafton, 2002). People with deficits in acquisition of implicitknowledge may need to rely on more effortful processes, suchas use of the goal-recursive algorithm, to make such movescorrectly. This effortful, elaborate processing could, in turn,interfere with acquisition of control routines like recognize-displace-compute (Sweller, 1988). Conversely, people with def-icits affecting how they acquire explicit knowledge may not beable to acquire the goal-recursive algorithm, for example, andthus may have to rely on implicit representations, which aloneare insufficient for correct performance. Performance deficitscould conceivably also stem from difficulties determining whenimplicit representations are insufficient and thus when to turn toexplicit knowledge instead.

These analyses, although speculative, suggest that it is crucialto identify what elementary processes are engaged on amoment-by-moment basis during complex performance. Thetradition has been to interpret Tower of Hanoi performance interms of a unitary ability to create and follow plans, which is acase of defining cognitive abilities in terms of tasks rather than

underlying mechanisms (see also Feinberg & Farah, 2006).However, there has been a shift toward analyzing Tower ofHanoi performance in terms of multiple elementary cognitiveprocesses, such as working memory (e.g., Goel, Pullara, &Grafman, 2001; Miyake, Friedman, Emerson, Witzki, & How-erter, 2000; Morris, Miotto, Feigenbaum, Bullock, & Polkey,1997; Numminen, Lehto, & Ruoppila, 2001). This shift hasdriven methodological changes in which coarse performancemeasures, such as total time per solution and total number ofmoves deviating from the optimal path (e.g., Owen, Doyon,Petrides, & Evans, 1996; Ward & Allport, 1997), have beenreplaced with move-by-move measures that allow better mea-surement of processes like memory retrieval (e.g., Anderson,Albert, & Fincham, 2005). The present study contributes to thisshift, focusing on perceptual and attentional processes and therole of abstract routines that organize cognitive operations andactions on the environment. The associated methodologicaldevelopment was to introduce the use of saccade-contingentdisplay changes to probe the contents of participants’ problemrepresentations.

Conclusions

In this study, we examined performance in a domain thattraditionally has been analyzed in terms of plans—sequences ofgoals stored in and retrieved from memory—and showed thatthey actually make incorrect behavioral predictions. We pro-pose that the functionality that has been attributed to plans inprevious models can come from relatively simple and abstractrule-based representations interacting with the environmentthrough perception and attention. This work takes a new step inan evolution of theorizing about goal-directed behavior that hasled from architectural constructs dedicated to goals (Andersonet al., 1993; Newell & Simon, 1972) to ordered sequences ofgoals represented in declarative memory (Anderson & Doug-lass, 2001), to sequencing of goal retrievals through environ-mental cues and cue-selection heuristics (Altmann & Trafton,2002), and now to the functionality of goals themselves beingreplaced by situated control routines. Whether plans can simi-larly be eliminated from other accounts of behavior tradition-ally viewed as planful remains to be seen. In the interim, wehave illustrated some implications of this work for interpreta-tion of everyday task performance and for neuropsychology,offering a reanalysis of a popular diagnostic task on the level ofmore elementary cognitive processes.

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Appendix

Computational Simulation of the Selective-Attention Model

Here we describe a computational simulation of the selective-attention model. The point of the implementation is to show thatthe mechanisms we posit are functionally sufficient to find theoptimal solution path in the Tower of Hanoi, as people are ableto once they have learned how, and that they do so in anempirically plausible way. In the following, we first describethe core mechanisms, at a level that is meant to help guide aneffort to replicate them in software, and then show fits toempirical response latencies from the control problems of Ex-periment 1. Executable and documented Common Lisp code forthe simulation and spreadsheets used for parameter estimationare available at msu.edu/�patsenka/sam

Model Mechanisms

Figure A1 shows a trace of the model running through thefirst 32 moves of a six-disk problem (there are 63 moves total).In this particular problem, all disks are on Peg A in the initialstate and on Peg C in the final state. The first column of thetrace shows the move number. The second column shows themove itself— on Move 1, for example, “1AB” means that Disk1 was moved from Peg A to Peg B—followed by the resultingstate, with the disks on a peg shown in parentheses and a pegwithout disks identified as “0.” The third column shows thesimulated latency for that move. The fourth column gives somedetails of the internal processing leading up to that move. Forexample, in the processing leading up to Move 4, the systemrecognized the new tower t83 that was completed on Move 3(Disks 1 and 2 on Peg C). Recognizing this new tower triggereddisplacing a disk from the old tower, specifically Disk 3 fromwhat remains of the initial tower t82 on Peg A.

The simulation’s performance is coordinated by therecognize-displace-compute routine, which has a number ofparameters that are bound by perceptual and attentional opera-tions. When these parameters are bound, the routine executes,which changes the world so as to create a new set of bindingsthat cause it to execute again, and so on. In the following, wecharacterize this routine in a “production rule” format and thendescribe it clause by clause:

recognize-displace-compute

when perception recognizes a new tower, then

displace a disk from the old tower to the empty peg,

apply goal-recursion to the new tower to find where to move Disk 1,

move Disk 1, move Disk 2 to the other peg, and move Disk 1 to Disk 2

Perception recognizes a new tower whenever Disk 1 is placed onDisk 2 (although the tower itself may extend to larger disks; seeFigure 2). This recognition is a signal to “displace” a disk from anold tower (old towers are described in more detail below) to theempty peg. The empty peg is effectively the peg that was clearedby assembling the new tower; if it is not physically empty, then itis subjectively empty (we would say), because any disks it holdsbelong to neither the new tower that was just recognized nor theold tower from which a disk is being displaced, so at this point,they are not selected for attention. When the disk from the oldtower is displaced, it remains selected for attention in its newlocation, indexed as the just-displaced disk. The goal-recursionalgorithm then starts, taking the just-displaced disk as the destina-tion for the largest disk of the new tower and recursing up the newtower until it determines the destination for Disk 1. Disk 1 is thenmoved to that destination, after which two additional moves aremade on the basis of heuristics carried over from previous models(e.g., Altmann & Trafton, 2002), and for which we found someevidence in Experiment 3 (in the three-move error sequences thatsome participants made repeatedly). Thus, recognize-displace-compute triggers a three-move sequence, the effect of which is totransfer a two-disk tower (Disks 1 and 2) from the top of the newtower to another peg.

The heuristics governing the second and third moves of thisthree-move sequence are not core assumptions of the model, whichfinds the same solution path if they are omitted such thatrecognize-displace-compute triggers only the first move of thesequence. The only processing differences in this alternative ver-sion are that recognize-displace-compute applies on every secondmove, rather than on every fourth move (as in the trace in FigureA1) and that to apply on the additional moves, it must considerDisk 1 by itself to be a “tower,” rather than requiring a tower toinclude at least Disk 2.


The parameters of the recognize-displace-compute routine arebound by attentional indexes. There is one such index for the newtower, one or two indexes for old towers, and one for the just-displaced disk. We interpret an index as a pointer to an object—either a tower, or the just-displaced disk. We assume that towerobjects in particular are disjoint, such that any disks included in anew tower when it is recognized lose their association with any oldtower. Thus, as an old tower is disassembled, it shrinks; when thelast disk of that tower is displaced, the pointer to that object isde-allocated, and that tower is no longer represented in the system.In the actual implementation, a tower is a unique symbol assignedto each disk in the tower (e.g., “t83” from earlier). A disk can beassigned at most one tower, so when a disk is assigned a newtower, one fewer disk is assigned the old tower (the tower“shrinks”). When a tower symbol is assigned to no disks, it isde-allocated (erased from the system).

We noted in passing above that there can sometimes be two oldtowers, rather than just one. An example occurs on Move 12 of thetrace in Figure A1. One old tower at this point is t84 on Peg B,which contains only one remaining disk (Disk 3). The other oldtower at this point is t82 on Peg A, which is the remnant of thetower from the initial state of the problem (Disks 5 and 6). On suchmoves, only one old tower (t84 in this case) can ever be displaced

from, because moves from the other old tower are always blockedin some way, so no additional heuristic knowledge is necessary toselect one old tower over another. However, we do assume thatonly the tower with the displaceable disk, which is always the mostrecent old tower and is also always the old tower with the smallerdisk, is indexed for purposes of computing the empty peg.

This implementation of indexes seems not to stretch availableestimates of indexes as a resource. The model needs at most fourindexes at once—the new tower, two old towers, and the just-displaced disk—and estimates from multiple object tracking stud-ies suggest there are four or five (e.g., Pylyshyn, 2000). Nonethe-less, the question of what limits there might be on availability ofindexes in routine behavior, in which the control representationsparameterized by indexes may be quite different and much richerthan in multiple-object tracking, would be a relevant question topursue in future studies.

Model Fits to Response Latencies

Figure A2 plots the simulated response latencies from Figure A1against empirical response latencies for the control problems ofExperiment 1. We estimated three parameter values to fit the data.

(Appendix continues)

SimulatedMove Move and latency

# resulting state (ms) Comments on processing during the move

1 1AB: (2 3 4 5 6) (1) 0 8410 Recognized t82 (1 2 3 4 5 6) on peg A. Applied goal-recursion to t82.2 2AC: (3 4 5 6) (1) (2) 12313 1BC: (3 4 5 6) 0 (1 2) 12314 3AB: (4 5 6) (3) (1 2) 1770 Recognized t83 (1 2) on peg C. Displaced disk 3 from t82.5 1CA: (1 4 5 6) (3) (2) 2559 Applied goal-recursion to t83.6 2CB: (1 4 5 6) (2 3) 0 12317 1AB: (4 5 6) (1 2 3) 0 12318 4AC: (5 6) (1 2 3) (4) 1770 Recognized t84 (1 2 3) on peg B. Displaced disk 4 from t82.9 1BC: (5 6) (2 3) (1 4) 3887 Applied goal-recursion to t84.10 2BA: (2 5 6) (3) (1 4) 123111 1CA: (1 2 5 6) (3) (4) 123112 3BC: (1 2 5 6) 0 (3 4) 1770 Recognized t85 (1 2) on peg A. Displaced disk 3 from t84.13 1AB: (2 5 6) (1) (3 4) 2559 Applied goal-recursion to t85.14 2AC: (5 6) (1) (2 3 4) 123115 1BC: (5 6) 0 (1 2 3 4) 123116 5AB: (6) (5) (1 2 3 4) 1770 Recognized t86 (1 2 3 4) on peg C. Displaced disk 5 from t82.17 1CA: (1 6) (5) (2 3 4) 5215 Applied goal-recursion to t86.18 2CB: (1 6) (2 5) (3 4) 123119 1AB: (6) (1 2 5) (3 4) 123120 3CA: (3 6) (1 2 5) (4) 1770 Recognized t87 (1 2) on peg B. Displaced disk 3 from t86.21 1BC: (3 6) (2 5) (1 4) 2559 Applied goal-recursion to t87.22 2BA: (2 3 6) (5) (1 4) 123123 1CA: (1 2 3 6) (5) (4) 123124 4CB: (1 2 3 6) (4 5) 0 1770 Recognized t88 (1 2 3) on peg A. Displaced disk 4 from t86.25 1AB: (2 3 6) (1 4 5) 0 3887 Applied goal-recursion to t88.26 2AC: (3 6) (1 4 5) (2) 123127 1BC: (3 6) (4 5) (1 2) 123128 3AB: (6) (3 4 5) (1 2) 1770 Recognized t89 (1 2) on peg C. Displaced disk 3 from t88.29 1CA: (1 6) (3 4 5) (2) 2559 Applied goal-recursion to t89.30 2CB: (1 6) (2 3 4 5) 0 123131 1AB: (6) (1 2 3 4 5) 0 123132 6AC: 0 (1 2 3 4 5) (6) 1770 Recognized t90 (1 2 3 4 5) on peg B. Displaced disk 6 from t82.

Figure A1. Trace output from the selective-attention simulation for the first 32 moves of a six-disk problem(which has 63 moves total). In the initial state (before Move 1), all disks were on Peg A, and in the final state,all disks will be on Peg C.


The first is the time for one recursion of the goal-recursive algo-rithm, estimated to be 1,328 ms. This time is charged to goal-recursion moves (1, 5, 9, . . .), multiplied in each case by thenumber of recursions needed for the tower on that move (five forMove 1, one for Move 5, two for Move 9, . . .). In terms of humanperformance, this time has to accommodate an eye movement to a

disk and then another eye movement to its target peg (see Figure 1for sample eye movements). The second parameter value is thetime to select and move a disk, estimated to be 1,231 ms. This timeis charged to every move, including goal-recursion moves, and interms of human performance has to accommodate a mouse move-ment to the to-be-moved disk followed by a mouse movement tothe target peg. The third parameter value was time to recognize anew tower and then attend to an old tower from which to displacea disk, estimated to be 539 ms. This time is charged to Move 1 andto all moves before goal-recursion moves (1, 4, 8, . . .).

This third parameter is the only one specific to the selective-attention model; the others have to be estimated for plan-basedmodels also. The estimate for this parameter, of about half asecond, seems in accord with the perceptual and attentional oper-ations we posit on these moves (recognizing a new tower andattending an old one, respectively), although this is at best aqualitative assessment. In more quantitative terms, excluding thisparameter degrades the fit of the model. With the parameter,root-mean-squared deviation is 402 ms; with only the first andsecond parameters, root-mean-squared deviation for the best fit is469 ms. A spreadsheet showing both fits, which is also what weused to estimate the parameters we then set in the simulation, isavailable with the other online materials at the previously men-tioned URL.

Received August 27, 2008Revision received November 3, 2009

Accepted November 4, 2009 �

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Figure A2. Simulated response latencies (Model) compared with empiricalresponse latencies (Data) from the first 32 moves of the control problemsof Experiment 1.


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