specifying executive representations and processes in ...specifying executive representations and...

Specifying executive representations andprocesses in number generation tasks

Sophie K. Scott and Philip J. BarnardMRC Cognition and Brain Sciences Unit, Cambridge, UK

Jon MayDepartment of Psychology, University of Sheffield, Sheffield, UK

The Interacting Cognitive Subsystems framework, ICS (Barnard, 1985) proposes that central ex-ecutive phenomena can be accounted for by two autonomous subsystems, which process differentforms of meaning: propositional and schematic (implicational) meanings. The apparent supervi-sory role of the executive arises from limitations on the exchange of information between theseand other cognitive subsystems. This general proposal is elaborated in four experiments in whicha total of 1,293 participants are asked to spontaneously generate a large verbal number to varyingtask constraints, with the intention of specifying the representations of number and task that un-derlie responses. Responses change systematically according to participants’ use of explicit prop-ositional information provided by the instructions, and inferred implicational information aboutwhat the experimenter is requesting. There was a high error rate (between 6% and 24%), partici-pants producing responses that did not fall within the large range indicated by the instructions.The studies support the distinction between propositional and implicational processing in execu-tive function, and provide a framework for understanding normal executive representations andprocesses.

Number generation has been utilized as a complex, “difficult” cognitive task in a variety ofways in psychology. For example, random number generation has been used to load the cen-tral executive of working memory; participants’ responses tend to be non-random and thisworsens with rate (Baddeley, 1966). This is not found when participants are required to pro-duce random key presses (Towse, 1998), indicating that when participants can make randomselections from a range of visual stimuli where the whole array of possible responses are visibleto be selected from, more random behaviour is possible. When responses must be generatedfrom a stored representation, then behaviour is less random and more vulnerable to disruptionwith increased load.

Requests for reprints should be sent to Sophie K. Scott, Institute of Cognitive Neuroscience, University CollegeLondon, Alexandra House, 17–19 Queen Square, London WC1N 3AR, UK. Email: [email protected]

We would like to thank David Golightly, Wendy Knightley, Jackie Andrade, Geraldine Owen, Jane Hutton, andTom Manly for help collecting data, and thank John Towse and an anonymous reviewer for valuable comments on anearlier version of this manuscript.

Ó 2001 The Experimental Psychology Societyhttp://www.tandf.co.uk/journals/pp/02724987.html DOI:10.1080/02724980042000327

THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 2001, 54A (3), 641–664

http://www.tandf.co.uk/journals/pp/02724987.html

Other simpler tasks can also reveal non-random behaviour. If asked to generate a numberbetween zero and nine, participants produce the digit seven at strikingly high rates (Dietz,1933). This result is robust enough to form the basis of magic tricks (Gardner, 1973). Initiallythis “heptaphilia” was assumed to be an “unmotivated preference” (Heywood, 1972). Laterstudies suggested that participants select seven because it is consistent with the required spon-taneity (Kubovy & Psotka, 1976), as it is not even, does not form part of the 3, 6, 9 sequence,and does not fall at the ends or middle of the 0–9 range. If the number seven is explicitly men-tioned, then the frequency of its generation drops significantly, as it does if the range ischanged. Further work showed that responses could be more subtly biased; if participants areasked to “think of a one-digit number”, the rate of “one” responses is increased and those of“seven” decreased (Kubovy, 1977). Replicating an effect initially described by Hsue (1948),when a “four-figure” number was requested the initial digit of “four” was much more fre-quent (27.4%) than if a number between 1,000 and 9,999 was requested (4.3%). The genera-tion of a candidate response is implicitly influenced by aspects of the task instruction that donot actually alter explicit aspects of the task.

An unusual aspect of all these studies is that all participants were asked only one question,many being stopped on campus and asked to think of a number, and others asked to “writedown the first number that comes to mind”. The demand characteristics of this “non-experi-mental” task and setting entail novelty and a lack of preparation. These tasks are therefore sim-ple, but have features of an executive task (Shallice, 1988). As in another simple numericexecutive task, cognitive estimates (Shallice & Evans, 1978), performance of this task requiresencoding what the boundaries are, forming a representation based on this of an appropriateresponse, checking this, then responding. Kubovy interprets the consistency in number gen-eration as reflecting an editing of an appropriate response (Hockett, 1967) after some primarygeneration. An alternative (and not necessarily contradictory) explanation is that there is aschematic representation of what would be an “appropriate” response, which is used to checkthe output, resulting in consistency across participants. This paradigm and response genera-tion framework will be used further to address and extend the specification of executive func-tion in number generation, and the role of underlying representations.

A framework for investigating executive function

Existing theoretical positions share a view of executive function as comprising high-level pro-cesses involved in the supervision, planning, and control of attention, especially in operationsthat are novel to a participant, and which have some non-routine, non-automatic properties.This paper will adopt the Interacting Cognitive Subsystems (ICS) framework in an attempt tomodel both process and representation in executive function. In contrast to the working mem-ory (Baddeley & Hitch, 1974) and the Supervisory Attentional System (Norman & Shallice,1986; Shallice, 1988) approaches, Barnard (1985) proposed a cognitive architecture that dis-pensed with the need for a controlling or supervisory subsystem. Instead, the flow of mentalrepresentations between nine independent, but interacting, cognitive subsystems is self-regulating.

The central concept is that different types of mental representation are required to accountfor the richness of human cognition, and that each type of representation is received, stored,and processed by a functionally independent subsystem. The flow of representations through

642 SCOTT, BARNARD, MAY

the overall architecture is data driven rather than being controlled or scheduled, with thereciprocal processing of the central engine serving to maintain and regulate the flow. Thereciprocal exchange of information between two particular subsystems, which deal with prop-ositional and implicational representations, respectively, has been given the name “centralengine”. This denotes its equivalent role to the central executive of the working-memorymodel, while trying to avoid the command-giving connotations of an executive component(Barnard, 1999; Barnard & Teasdale, 1991; Teasdale & Barnard, 1993, pp. 81–82).

Complex novel tasks require many exchanges of representations between subsystems as theinformation is elaborated and evaluated in terms of memory records. The central engine con-tinually compares specific propositional information about the task (e.g., the goal to beachieved, the last action completed, and the outcome of that action) with generic implicationalinformation distilled from the performance of related tasks in the past.

The propositional level of representation embodies semantic knowledge about entities inthe world, and their specific attributes, identities, and interrelationships, whereas theimplicational level embodies more generic, schematic information about propositions, includ-ing affective information. Teasdale and Barnard (1993) describe these two forms of mentalrepresentations as two different levels of meaning. Propositional representations are of cold,factual knowledge whereas implicational representations are of feelings and implicit knowl-edge. The central engine exists because the two subsystems that receive these representationsprocess them, to produce new implicational and propositional representations, respectively.In consequence the two levels of meaning are continually influencing each other.

When the ICS framework is applied to the “encode task, generate response, checkresponse, and produce output” model of single-number generation then it becomes possible toexpand on this approach. The distinction between propositional and implicational meaningwithin the ICS account facilitates an investigation of different components of this executivetask in a way that other models do not. In addition to the encoding of the boundary numbers(e.g., “a number between one and ten”) as a propositional representation, there will also be amore schematic representation of the task; generating a response will involve both proposi-tional and implicational features of the candidate response. Such a request is not simply aproposition, it is a request for information that must be interpreted to determine what kind ofnumber is required by the experimenter—in Kubovy’s terms, what response is “appropriate”.Thus both how the task is encoded, and subsequent formation of propositional andimplicational representations of a response based on this will result in an appropriate numberbeing generated, and also the accuracy of the checking, before output is produced. An existingtheory of spoken number generation (McCloskey, Sokol, & Goodman, 1986) uses a syntacticnumber frame to parse output into base ten number forms; this account cannot address whatthe content of this number frame might be nor how it is specified. The use of abstract sche-matic models of number and task proposed by the ICS framework, in contrast, predicts consis-tency across responses. This is congruent with Kubovy’s (1977) and Hsue’s (1948) data—forconsistency in responses to arise, participants must be generating a number to task constraintsthat they are themselves representing, rather than retrieving randomly. The implication isthat humans do not “throw mental dice”—generative behaviour will be influenced byimplicational and propositional representations of the task.

ICS is a framework, rather than a theory, and does not therefore make specific predictionsabout the exact representations used. Instead, ICS predicts that manipulation of the

EXECUTIVE FUNCTION IN NUMBER GENERATION 643

propositional form of the question will imply differential salience of these generic number fea-tures, which can be identified in the responses. Analysis of the numbers generated to differentquestions enables explicit investigation of specific strategies and representations that underliethe generated answer, revealing the schematic models of number used by the participants.This leads to the following hypothesis about number generation:

ICS predicts that there will be consistencies in responses, across participants. These will bebased on schematic models of number encoded at an implicational level of meaning, withsystematic combinations of generic features such as complexity and magnitude. The precisecombinations, and thus models, can be determined from the data collected. These consisten-cies will be further influenced by propositional and implicational features of the taskinstructions.

This is not a test of ICS; as a framework, it cannot be disproved. However, if the results runcontrary to these hypotheses, then central features of the ICS architecture would not betenable. It is also important to establish that these predictions cover many factors similar tothose addressed by Kubovy (1977) and Hsue (1948), and the aim is not to contradict these, butto integrate them into a general cognitive architecture.

Following on from the studies where participants generated numbers to constraints,(Hsue, 1948; Kubovy, 1977; Kubovy & Psotka, 1976), Experiment 1 was designed to establishhow strategies for the generation of large numbers are influenced by implicational propertiesof the question, while maintaining a similar overall propositional magnitude range. Experi-ments 2, 3, and 4 examined the effects of different manipulations on the questions, affectingthe parity, magnitude, and complexity of the boundaries provided. In all four experiments, thequestions and responses were verbal; this minimized the information available to participants,and thus their responses were based on their representation of the task. Written questionsand/or responses would facilitate extra checking of the response against the criteria, and as inthe random key press experiments of Towse (1998), result in responses that are “morerandom”.

Basic design

The four experiments described in this paper all follow the same general design. As inKobovy’s studies, large numbers of naïve participants were approached in informal settings atuniversities and research units and asked brief questions. The first question was to “think of acolour name”, a deliberately simple request to reduce participants’ anxiety at being tested.The second question was the number generation task. If participants asked for clarification,the question was repeated. Each participant was tested only once and excluded if they hadoverheard others being tested. No personal information was requested, nor were participantsinformed if their response was incorrect. All responses were recorded and the participantswere debriefed immediately afterwards. Seven experimenters did the testing, following thesame script. Throughout testing, different questions were asked in rotation, to avoid order orcohort effects.


EXPERIMENT 1

Method

Participants

A total of 520 people were approached and invited to participate in a brief psychological study thatrequired no personal information.

Design

In this first study, participants were asked to generate a number that lay in the range one million to tenmillion. There were four separate conditions, in which the aspects of the task demands were changed, in away that did not affect the overall range of possible responses. In the one–ten condition, participants (n =136) were asked “please could you give me a number between one million and ten million”. In the ninescondition, participants (n = 128) were asked “please could you give me a number between one millionand nine million, nine hundred and ninety-nine thousand, nine hundred and ninety nine”. In the seven-figure condition (n = 129), participants were asked “please could you give me a seven-figure number—that’s a number between one million and ten million”. In the random condition (n = 127), participantswere asked “please could you give me a random number between one million and ten million”.

The basic hypothesis is that variations in the form of the question posed will change the way that par-ticipants derive an answer. The following analyses identify whether a response is selected at randomfrom the full possible range, or if there are systematic regularities occurring in responses.

Results

Responses were predominantly verbal numerals. Seven participants produced digit strings, allin the seven-figure condition, and of these three were incorrect. The characteristics of theresponses produced are summarized in Appendix A. The distribution of responses is skewedtowards the lower boundary. The existence of modal responses indicates consistency; forexample in the one–ten condition, the chance of 20 identical numbers being randomly selectedis less than 10–140.

A large number of participants made overt errors, answering with a number that lay outsidethe specified range. The error responses will be further analysed at the end of this paper. Allcorrect responses were analysed to investigate consistency. The first analysis examined theelaboration of numeric detail, the second examined the distribution of initial digits, and thethird specifically examined parity and position in elaborated responses.

Elaboration of numeric detail

Table 1 illustrates the four basic categories of numeric elaboration, which renders the num-ber of base 10 “slots” equivalent in the thousands and hundreds-unit ranges. Figure 1 showsthe percentage of all correct responses in each class, according to the question the participantshad been asked. Significant variation in this distribution and later experiments was assessedusing chi-square tests. For the number structure data, the question conditions formed therows of a contingency table, with the categories of answer forming the columns.

There was significant variation in the amount and type of elaboration specified in responseto the four questions, c2(9) = 109, p < .05. In the one–ten condition 66% of correct responseswere unelaborated round millions. In the seven-figure condition, numbers elaborated in both


the thousands and hundreds-units ranges dominated the correct response distribution. In therandom condition participants produced either a round million response or a response elabo-rated in the hundreds-units range, together accounting for more than 80% of the correctresponses. In the nines condition 64% of the correct responses were elaborated in either thethousands and hundreds-units ranges or in the hundreds-units range alone, when the modalresponse of 1,000,001 accounted for nearly half of all the responses. Responses consisting ofmillions and thousands only were rare throughout.

The number of millions in unelaborated and elaborated responses

Figure 2 shows the distribution of initial digits—that is, those used to mark the number ofmillions for the correct responses. The data are presented separately for unelaborated roundmillion responses and for all those responses that were elaborated in some way. The distribu-tion of initial digits in each condition, by response type, was tested with chi-square tests. Sepa-rate contingency tables were constructed for each response type, in each condition, with theinitial digits forming the columns, as there were varying numbers of each response type in eachcondition. From Figure 2, participants marked unelaborated responses in a quite differentfashion from responses that were elaborated, and the distribution is significantly differentfrom chance in the one–ten and random conditions, c2(8) = 22.42 and 24.8, respectively, p <


TABLE 1Classification of the degrees of elaboration of numbers generated

Category Examples

[millions] [—] [—] 1,000,000 5,000,000 10,000,000[millions] [thousands] [—] 1,100,000 5,124,000 9,009,000[millions] [—] [hundreds-units] 1,000,001 5,000,378 9,000,900[millions] [thousands] [hundreds-units] 1,001,001 5,124,378 9,999,999

Figure 1. Percentages of correct responses generated in each number elaboration category, according to questiontype in Experiment 1.

.05. This is due to an increase in the percentages of the two odd digits in the lower part of themagnitude range (3 and 5). In the elaborated responses, the distribution was significantly dif-ferent from chance for all conditions; for the one–ten, seven-figure, nines, and random condi-tions, c2(8) = 74.2, 65.3, 80, and 207, respectively, p < .05. This is due to an elevated frequencyof the initial digit 1 (119 of all 259 elaborated responses). If the contingency tables are repeatedwithout the digit 1, none of the distributions was significant.

Parity and place in elaborated responses

The next analysis examines the parity of digits produced in the elaborated responses. Inthese the digit at the start of the elaboration is termed the “head”, and the digit at the end the“tail”. The parity of the head and tail elements of elaborated numbers is shown in Appendix B.Instances of single-digit elaboration are indicated separately. Neither the head nor the tail ofextensive elaborations deviated significantly from chance in their parity. However, the single-digit responses are predominantly odd, differing significantly from chance, for three of thefour conditions (there were only two of these responses in the seven-figure condition).


Figure 2. The digit used to specify the “million” slot of participant’s responses in Experiment 1, expressed as a per-centage of correct responses. Upper panel represents elaborated responses, lower panel unelaborated responses. Up-per boundary responses (“ten million”) are not included, as these could not be elaborated without also making anerror.

Discussion

Following on from previous studies (Kubovy, 1977; Kubovy & Psotka, 1976), the results ofExperiment 1 show consistency in the way participants generated large numbers in responseto different questions. Participants’ responses can be characterized in terms of the extent ofnumeric elaboration and systematically marking (or not) of parity in some positions. Somecombinations occurred with considerable frequency, other combinations were rare. In addi-tion, there was an influence of the question posed, and although this was a simple task, therewere also many errors.

The thematic structure of the numbers generated can be used to draw together the patternsseen in responses. Using the terminology of systemic grammar (e.g., Halliday, 1970), the firstelement of a number can be referred to as the “psychological subject” which defines the mag-nitude range (e.g., one million). The “psychological predicate” involves associated informa-tion, such as whether numeric detail is elaborated or not, and the degree of elaboration. Thus around number of millions constitutes a psychological subject with no psychological predicate.A number with full elaboration of all the number slots represents a psychological subject withpredicate elaboration. These two basic thematic frameworks are contrasted in Figure 3 (verbalnumerals Types A, B1, and B2) and distinguished from a third strategy of counting off a stringof digits (Type C, seen only in the seven-figure condition). A syntactic approach to numbergeneration (McCloskey et al., 1986) would only distinguish between verbal numbers and digitstrings. The motivation for using the semantic systemic approach is that it captures attributesof the semantic properties of the numbers, can be used to describe both implicational andpropositional structure, and thus may give a more explicit specification of the role of these rep-resentations in central executive function.

The distinctions drawn along thematic lines can be used to consider the propositional andimplicational underpinnings of particular responses. So, for example, A in Figure 3 might berepresented by a low, odd psychological subject (as seen in the one-ten condition); B2 by mini-mal elaboration of a single odd predicate (as seen in the nines condition); and B1 by


Figure 3. Different thematic structural frameworks for implicational models of large numbers.

complexity, that is, extensive elaboration of numeric detail in a predicate not specificallymarked for parity (as seen in the seven-figure condition). In the last two models, the psycho-logical subject was most likely to be one million. The dominant use of one million as thepsychological subject of number with some numeric elaboration in the psychological predicatecould arise for a number of reasons; overt marking of the lowest number of odd parity, ageneric influence of primacy, or thematic continuity (one million was the lower boundaryspecified in all four variants of the question posed). Alternatively, it could reflect an influenceof frequency and familiarity. Benford’s Law (Benford, 1938) demonstrates that low numbersare more frequent than higher numbers in the environment, particularly the number one. Inorder to clarify this particular issue, Experiment 2 was conducted where the lower boundary inthe questions was raised from one million to two million.

EXPERIMENT 2

Method

Two changes were made to the questions in Experiment 2. First, the lower boundary was raised to twomillion. This had two consequences. The parity of the initial boundary was now even, and so any influ-ence of parity in the instructions should affect responses in an opposite direction to that found in Experi-ment 1. Any bias towards beginning numbers with one million for some other reason would now result inan error. Second, no random condition was used, as responses in this condition were falling into two dif-ferent types. The procedure for sampling participants, asking the questions, and recording theirresponses was the same as that in Experiment 1.

Participants

A total of 335 participants were approached in coffee bars and sites on Nottingham University cam-pus and asked to participate in a quick experiment.

Design

There were three conditions, in which a number between two million and ten million was requested.In the two–ten condition (n = 115), participants were asked, “please could you give me a number betweentwo million and ten million”. In the nines condition (n = 110) participants were asked, “please could yougive me a number between two million and nine million, nine hundred and ninety-nine thousand, ninehundred and ninety nine”. In the seven-figure condition (n = 110), participants were asked “please couldyou give me a seven-figure number between two million and ten million”.

Results

Participants’ responses are summarized in Appendix A. As in Experiment 1, the responses areskewed towards the lower boundary, and the error rate is high, at 15% overall. Eight partici-pants now refused to produce a number. The modal value of the nines and seven-figure condi-tions is six million, and that of the two–ten condition, as in the one–ten condition, is fivemillion.


Elaboration of numeric detail

Once again, the structure of the responses varied according to the condition, as shown inFigure 4, and this variation was significantly different from chance, c2(6) = 219, p < .05.

As in Experiment 1, the dominant response was to produce a round million response in thetwo–ten condition. Although this was also a common response in the other two conditions, themost popular response in the nines condition was to produce a number elaborated in the hun-dreds-units range, again following the pattern of Experiment 1. Unlike Experiment 1, the 7-figure condition did not lead to a dominant response with a fully elaborated predicate.

The number of millions in unelaborated and elaborated responses

As before, the initial “millions” digit of the correct and boundary responses, correspondingto the psychological subject of the responses, was analysed for those who had produced roundmillion and elaborated responses in each condition (Figure 5).

The pattern of responses varies significantly from chance: elaborated responses c2(7) =26.22, 94.11, and 38.00, p < .05; unelaborated responses, c2(7) = 39.66, 22.42, and 15.97, p <.05, and is similar to that found in Experiment 1, with one exception. In Experiment 1 the ini-tial boundary of one million was predominantly repeated by participants who went on to elab-orate their response. The direct equivalent of this—that is, a dominance of elaboratedresponses starting “two million”—was reduced in all three conditions with more participantsstarting their response with “three million”. This pattern varies with condition; in the two–tencondition 36% of the total correct elaborated responses begin with the digit 2 and if the chi-square test is repeated without 2 it is not significant. However, in the nines and the seven-fig-ure conditions if the test is repeated without 2 the distribution is still significantly differentfrom chance, c2(6) = 53.8 and 31.5, respectively, p < .05. Thus the elaborated responses tend



to start with 2 but those starting with 3 are also elevated in frequency; in the seven-figure con-dition the initial digit 3 is more frequent than 2.

The unelaborated responses are characterized by low, odd initial digits in the two–ten andnines conditions (3 and 5). In the seven-figure condition the responses show a preference forthe odd numbers, 3, 5, and 7, but the modal response is 6, accounting for 24% of unelaboratedresponses.


As in Experiment 1, the parity of the head and tail of complex predicates and of simplesingle-digit predicates was assessed, as shown in Appendix B. Where there are single digitsspecified as predicates these are still predominantly odd, even when the parity of the initialpsychological subject is even. This is statistically significant in the nines condition, where themodal response is 2,000,001, not 2,000,0002. In more elaborated propositional predicates oddand even digits do not differ significantly in frequency.

Discussion

As found previously, responses with predicate elaboration were associated with the use of thelower boundary number as the psychological subject. This was not because this was the onlyboundary that could be elaborated correctly, as in the nines condition elaborated responsesbeginning nine million were uncommon. Use of the lower boundary to start an elaborated


Figure 5. The digit used to specify the “million” slot of participant’s responses in Experiment 2, expressed as a per-centage of correct responses. Upper panel represents elaborated responses, lower panel unelaborated responses.There were no upper boundary responses.

response was reduced, compared to Experiment 1, and the psychological subject “three mil-lion” was used more often. This seems to reflect a generic tendency towards odd numbers,despite even number boundaries.

The use of Implicational Model A again produces round numbers with an odd psycho-logical subject, with the exception of the increased mid-range use of six million in the ninescondition. Implicational Model C (digit strings) was again only used in the seven-figure condi-tion, and was incorrectly applied on 6 out of 11 occasions. Regardless of the parity of thepsychological subject, single-digit predicates (Model B2) are marked as odd.

In these two experiments the magnitude range was kept roughly equivalent, because thelarger a response is, the more complex a fully elaborated predicate must be. In addition largernumbers are poorly represented cognitively, and this worsens with magnitude (Poulton,1979). Experiment 3 was designed to examine the effects of a shift in magnitude range tosmaller numbers, thus increasing propositional knowledge about the numbers, and reducingthe potential complexity of a fully elaborated response. The number range chosen, betweenone and ten thousand, is the same as that used by Kubovy (1977).

EXPERIMENT 3

Method

Participants

A total of 319 participants were approached in coffee bars and sites on Nottingham University cam-pus and asked to participate in a quick experiment.

Design

The number range was between one thousand and ten thousand, and there were three separate ques-tion conditions. In the one–ten condition (n = 108), participants were asked, “please could you give me anumber between one thousand and ten thousand”. In the nines condition (n = 107), participants wereasked, “please could you give me a number between one thousand and nine thousand, nine hundred andninety nine”. In the four-figure condition (n = 104), participants were asked, “please could you give me afour-figure number—that’s a number between one thousand and ten thousand”.

Results

Participants’ responses are summarized in Appendix A. The error rate with these instructionswas 10% overall, lower than in Experiments 1 and 2. The distribution of responses is againskewed, with a bias towards responses nearer the lower boundary.

The elaboration of these responses cannot be analysed as in the previous experiments,because of the change in range. Instead, a breakdown of use of the four base 10 values of thenumbers produced for correct responses is shown in Figure 6. In terms of structure, the“thousand” category corresponds to the “million” category of the previous experiments, the“thousand, hundreds, tens and units” category corresponds to the “millions, thousands,units” or fully elaborated category, and the remaining two categories correspond to partiallyelaborated categories. Responses containing just a thousand and a single unit are identifiedseparately from all “other” combinations (i.e., some other predicate elaboration).


In the one–ten condition, there was a tendency for participants to provide an unelaboratedpsychological subject, although less frequently than in Experiments 1 and 2. Of the correctresponses, 18% of participants produced a response with a thousand and a single unit, such as1,001, and 32% produced more complex elaborations. Relatively few (12%) used all fournumber slots in their response. In the nines condition responses were spread across all cate-gories, whereas in the four-figure condition, the majority of participants fully elaborated thepossible four-figure complexity (57% of correct responses, with 24% producing digit strings).This distribution is significantly different from chance, c2(6) = 54, p < .05.

Figure 7 shows the distribution of digits used to specify the “thousand” value of the correctresponses. As in Experiment 1, participants who elaborated their response generally beganwith “one thousand” although the predominance of this response was lower in this experimentthan in Experiment 1, and 2–6 are equally represented as initial digits for the elaboratedresponses. Those who made an unelaborated response still produced the mid-range “5,000”.The distributions of initial digits in both elaborated and unelaborated conditions varied signi-ficantly from chance, c2(8) = 98, 99, and 26 for elaborated and 21, 15.6, and 21 forunelaborated responses, respectively, p < .05. If the chi-square tests are repeated without 1 theone–ten and nines distributions are still significant, c2(8) = 17.06 and 20.09, respectively, p <.05, reflecting the increased use of other lower numbers as mentioned earlier.


The parity of the digits used at the head and tail of elaborated predicates, and single digits,is shown in Appendix B. This again shows that if the predicate is elaborated, but consists of asingle digit, that digit is typically odd (for the one–ten and nines conditions), whereas morefully elaborated predicates are equally distributed between even and odd digits.



Discussion

The patterns of response with the smaller numbers are similar to those found in the previousexperiments. The differences can be attributed to a better propositional representation of themagnitudes concerned; magnitude is less complex to encode from the verbal instructions, andit is easier to produce a fully elaborated number. This may explain the increased use ofelaborated responses except where a lack of elaboration is cued by the instructions (i.e., in theone–ten condition). In addition the Model C digit string responses were very frequent in thefour-figure instructions (21 occurrences), and these were all correct. In the 2 seven-figure con-ditions participants produced the wrong number of digits on nearly 50% of occasions. It maythus be easier to produce four-digit strings, possibly due to participants not needing to countto achieve a numerosity of four.

In producing elaborated responses, the lower boundary of the task instructions is stillmainly used as the psychological subject of the response. In contrast to the data reported byKubovy (1977) there was no elevation of 4 as an initial digit in the four-figure condition. Thedifferent structure of the instructions may have affected this; Kubovy asked for “the first fourfigure number that comes to mind” (p. 362), and our request was for “a four-figure number—that’s a number between one thousand and ten thousand”. In thematic terms, the phrase “fourfigure” was the psychological subject of both instructions, but in the current study the psycho-logical predicate of the instructions described the number range, thus making it available in aproposition form for use in constructing a response.


Figure 7. The digit used to specify the “thousand” slot of participant’s responses in Experiment 3, expressed as apercentage of correct responses. Upper panel represents elaborated responses, lower panel unelaborated responses.Upper boundary responses (“ten thousand”) are not included, as these could not be elaborated without also making anerror.

This adoption of the lower boundary as a “default” psychological subject in the responsehas also been noted in all three experiments. It is a successful strategy, as an elaboration of aresponse starting with the lower boundary must be correct. It may also reflect a primacy effect,the participants forming their response as soon as they hear the first boundary, and not beinginfluenced by the latter boundary. The lower boundary may simply form the psychologicalsubject of the task, and as such be used as the psychological subject of the response. To exam-ine this, Experiment 4 used a more complex lower boundary that could not simply be repeated.

EXPERIMENT 4

Method

This final supplementary experiment uses the same range of possible responses as Experiment 1,with “ten million” as the upper boundary, but with “nine hundred and ninety-nine thousand, nine hun-dred and ninety nine” used as the lower boundary instead of “one million”. The psychological subject ofthis lower boundary, “nine hundred . . .”, cannot be used as the psychological subjects of a correctresponse. The propositional meaning of this number (one less than a million) must be represented cor-rectly (specifically or generally) for a correct response to be produced.

The question asked in this experiment was also a variant of the nines condition in Experiment 1, as itis of equivalent overall complexity. Thus the question “please could you give me a number between ninehundred and ninety-nine thousand, nine hundred and ninety nine and ten million” was asked by twoexperimenters. There were 119 participants tested in the same manner as in Experiments 1 to 3.

Results

The responses of the participants in this experiment are summarized in Appendix A. Strik-ingly, 26% of the participants fail the task and one in eight participants report it to be impossi-ble. In 90% of the incorrect responses, participants produce a response below the lowerboundary. The modal response was the upper boundary. Of the correct responses, the major-ity (n = 49, 64%) were unelaborated psychological subjects, as shown in Figure 8. This con-trasts with 28% unelaborated responses in the nines condition of Experiment 1. No other



category accounted for more than 20% of correct responses. This distribution is significantlydifferent from chance, c2(3) = 64, p < .05.

Figure 9 shows the identity of the “millions” digit of the correct responses, and can becompared with the nines condition in Figure 2. Unlike this condition, there was a low rate ofelaborated responses starting with the digit 1, which occurred at the same rate as 2 and higherrates for the initial digit 9. In the unelaborated responses the boundary response “ten million”was the modal response, and one and nine million are also elevated. The shape of these distri-butions indicates a preference for the boundaries of the range, with the mid-point also beingused by those who do not elaborate their responses. The overall distribution of both elaboratedand unelaborated responses varied significantly from chance: elaborated, c2(8) = 16;unelaborated, c2(8) = 19; both p < .05. If the elaborated analysis is repeated without 1, the dis-tribution is significantly different from chance, c2(8) = 15.81, p < .05, reflecting the increaseduse of 2 and 9.

Discussion

The high rate of boundary responses (11%), which has been at 1.5–5% in previous experi-ments, coupled with the “refusals”, suggests that participants considered this to be a trickquestion, many apparently misrepresenting the task as being for a number between 9,999,999and 10,000,000. There were many errors that went too low.

In Experiment 1, the nines condition had a question that was equivalent in its gross com-plexity, mentioning a round million as one boundary and an elaborate number composed ofnines as the other boundary, and yet this produced an error rate of only 13% and resulted prin-cipally in elaborated responses beginning “one million . . .”. The difficulties our participantsexperienced in answering the question in this experiment are not due to the complexity of thequestion; it is the structure of the complexity that is important. If they adopt the implicationalmodel A, to produce an unelaborated number, they no longer distribute their choice of firstdigit towards lower, odd numbers. A preference for “nine million . . .” can be seen as the avoid-ance of the complicated lower boundary; this may be analogous to the avoidance of the


Figure 9. The digit used to specify the “million” slot of participant’s responses in Experiment 4, expressed as a per-centage of correct responses. Solid lines represent elaborated responses, dashed lines unelaborated responses. Upperboundary responses (n = 12, 16%) are not shown since these could not be elaborated without also making an error.

complicated “nine million nine hundred and ninety nine” boundary in Experiments 1 and 2. Apreference for “one million . . .” indicates that these participants have recognized that thelower boundary is just below one million. The use of the mid-range “five million . . .” is con-gruent with the modal appearance of this mid-range number in other conditions. The initialboundary and the latter boundary have a different status in the participants’ interpretation ofthe task. If the psychological subject of the task is hard to represent, the task becomes more dif-ficult, and the predicate information must be used.

The full error corpus

In all these experiments the required response is precisely defined in the instructions, andthe range of possible responses is large. Nevertheless, 14% of 1,293 participants gave incorrectresponses, not including boundary responses or refusals. Overall there are far more errors thatgo too low than go too high (128 vs. 52). Table 2 shows the percentage of incorrect responses inall conditions.

The error rate varies significantly across conditions, c2(10) = 37.6, p < .05, principally dueto errors that fall below the lower boundary. The largest error rate occurred in Experiment 4 (anines question), and the seven-figure question in Experiment 2 was also error-prone. Smallerthan expected frequencies of errors occurred in the nines question of Experiments 1 and 2, andin the four-figure condition of Experiment 3. Thus there is no simple relationship betweenerrors and complexity of instructions, suggesting that participants were not simply misunder-standing the instructions.

The precise cause of this error variation is necessarily speculative, but certain patterns areapparent. In verbal numbers, errors occur at the start of a response, that is, in the psychologicalsubject. In Experiment 4 over two thirds of the low errors start with the psychological subject“nine hundred”, suggesting that many errors are resulting from attempts to repeat the lowerboundary of the question to start the response. The implicational model B2 could only lead toan incorrect response in this condition. The seven-figure question of Experiment 2 was theonly one where the two constraints in the question were not fully compatible (not all seven-fig-ure digits fall between two and ten million), and misrepresentations of the task may have raisedthe error rates. The overall error rate was lower in Experiment 3, consistent with the smallermagnitude range, but there are still differences across conditions. The successful use of non-syntactic digit strings for the four-figure question accounts for more correct responses.


TABLE 2Summary of error rates across all four studies, expressed as percentage of

all responses

Experiment One–ten Nines Four/seven-figure Random Overall

1 13% 8% 16% 18% 11%2 16% 5% 24% 15%3 11% 16% 4% 10%4 26% 26%

Overall 13% 14% 15% 18% 14%

Errors have a thematic structure; the above-range responses are characterized byunelaborated millions, and below-range responses by numbers elaborated with some predi-cate detail. This is shown in Table 3 for the above- and below-range responses. The below-range responses also include single-digit responses, which do not have a comparable thematicstructure.

GENERAL DISCUSSION

When generating large numbers in answer to a question specifying upper and lower bound-aries, participants do not select a number at random. Instead they make use of a few schematicmodels of the general properties of the to-be-generated number (shown in Figure 3), whichconstrain the numeric content of the propositional representation and hence response. Thefeatures of these implicational representations are magnitude, parity, and complexity. Theycan be determined by analysing the thematic structure of the responses.

Unlike the Kubovy studies, seven rarely formed the psychological subject. The initialdigit, like the number structure, was influenced instead by the condition, and the main fea-tures marking this were parity and magnitude. When range boundaries were specified withoutelaboration (e.g., one–ten; two–ten) the predominant response was to produce a round num-ber of millions—implicational model Type A. The psychological subject of the response wasmore likely to be odd than even, and it was most often drawn from the lower part of the magni-tude range (three or five). When elaborated responses were produced, if the lower boundarywas either one thousand or one million, then the most frequent response was to form an elabo-rated response with a psychological subject of one thousand or one million. If the lower bound-ary was raised to “two million”, then the percentage of responses that repeat the lowerboundary as the psychological subject was halved, with a corresponding increase in the use ofthe low-magnitude odd digit “three”. Thus the bias in elaborated responses is to begin with anodd number. When predicate detail is elaborated extensively, parity is never explicitly markedat either the head or the tail of the numeric elaboration. However, if predicate information iselaborated minimally, the final digit, like the psychological subject of the number, is markedwith odd parity.


TABLE 3The frequency of error responses of each response type that

were below or above the required range

Response type Frequency

Below range Single digits 15Responses with predicate elaboration 84Responses with no predicate elaboration 23Digit string 6Total 128

Above range Predicate elaboration 12No predicate elaboration 36Digit string 4Total 52

If the question posed refers to a number of digits, an increase in fully elaborated responseswas seen—that is, an increase in the proportion of responses that are marked for complexity.If the upper boundary specified in the question is itself complex, as in the nines conditions,then the elaboration is typically a single odd digit as the predicate. If the lower boundary men-tioned in the question is complex, responses typically are round millions, and the distributionof the psychological subjects alters with some marking the upper end of the target range aswell as the beginning or middle. Thus complexity in responses is marked differently acrossconditions.

The particular generic features of parity, magnitude, and complexity thus occur in particu-lar combinatorial forms, whereas other combinations rarely occur. These features recurthroughout the literature on different facets of number processing—for example, magnitudeand the problem size effect (Ashcraft & Battaglia, 1978; Groen & Parkman, 1972), parity andperceived similarity (Shepard, Kilpatrick, & Cunningham, 1975), and the role of complexitywhen learning the rules of concatenation and overwriting (Power & Dal Martello, 1990; Power& Longuet Higgins, 1978). This supports the relevance of the generic nature of the relevantunderlying representations, and their role in cognitive processes.

Systematic patterns also appeared within the unexpectedly high rate of errors. When par-ticipants erred by going below the lower boundary, most of the errors specified some numericdetail in the predicate, and few responses had no predicate elaboration. When participantserred by going too high, this pattern reversed, and most responses were round unelaboratedmillions. The large unelaborated high errors are similar to implicational model Type A; thesmaller, detailed predicate low errors are similar to model Type B. In each case, the anchoringeffect of the propositional lower boundary of the task has been forfeited.

Theories of central executive function that simply specify general resources for the super-vision, planning, and control of attention cannot explain either the distribution ofimplicational features used to generate responses or the close association of particular modelswith errors than go either too high or too low. This is the case even where specific reference tothe use of schema is made (e.g., Shallice, 1988) as the influence is based on the content andstructure of the schema in question. We propose that when participants interpret the questionposed, they generate a propositional interpretation of its content, and this, in turn, constrainsthe generation of a more abstract model of the target number, which is then used to generate apropositional representation of the target number in order to generate the surface structureform of the response. The target number may be evaluated for errors prior to responding(Barnard, 1999; Teasdale & Barnard, 1993).

The role of distinct models of number in the responses provides evidence for the involve-ment of schematic representations in the generation of large numbers, not explicitly entailedby question content alone. The underlying propositional content of the question posed musthave been interpreted to generate a new, more abstract specification, of the number to be gen-erated, an “appropriate number” in Kubovy’s terms. Response content comes from thisderived model and not from an interpretation of literal question meaning.

As schematic models of conceptual material can always be described in propositional terms,we cannot prove that the schematic models are necessarily encoded at a different level of repre-sentation. The pattern of data presented here is consistent with the hypothesis that two levelsof meaning are central to executive function, and with the idea that there is a close relationshipbetween representational content and process action. They also provide a context for the


development of a more detailed specification of how process action on representations cansubserve executive functions.

This apparently simple task gave rise to surprising numbers of errors. We suggest that theinferred models may themselves be influencing process action and its representational prod-ucts. If generically encoded information is structured to bring one attribute of a model intofocus, and this is reflected in the form of the final number generated, then the task demandsmay be such that many participants output this number directly without evaluating its contentagainst the original “propositional” representation of the instruction.

For example, when the question marks the boundary in a complex form, it may make thespecific identity of one boundary salient, and boundaries figure in the framing of the majorityof responses. Thus in the nines conditions in Experiments 1, 2, and 4, many participantsrepeated the lower boundary as the psychological subject of the number generated, with asimple, odd number forming the predicate, causing many errors in Experiment 4.

When the question marks the number of digits or the concept of randomness, or neither,the feature of boundaries is rendered less salient, and the feature of complexity may be morerelevant. When participants adopt the round number of millions or thousands strategy, theyare far more likely to go too high than too low when they err. This is consistent with the sugges-tion that they are primarily attempting to mark magnitude when going high or complexitywhen going low, rather than simply forgetting the boundaries.

The error rate also implies that specification of “normal” processing activity in this one-offnovel task may well involve a rather high proportion of responses being generated without aprocessing stage prior to output, in which model content is explicitly evaluated against thepropositional representation of the question. Had such a stage been routinely involved, as wasoriginally envisaged and as other frameworks would predict, error rates would have beenminimal.

This failure of explicitly evaluate responses against the task requirements can be seen inother tasks. For example, in the Moses illusion (Erickson & Mattson, 1981), participants fre-quently respond “two” if asked a question such as “How many animals of each type did Mosestake into the ark?”. However, if Moses is substituted with Nixon, error rates drop dramati-cally. Both Moses and Noah fit the same generic schematic model, unlike Nixon. Just as thequestion focuses on a particular attribute, in this case also a number, so the responses gener-ated here also appear to be bringing a particular attribute into sharp focus, whilst perhaps sub-jecting other attributes of their response to an attenuated evaluation.

Conclusion

The detailed distribution of correct responses fits an explanation of central executive func-tioning based upon a model of reciprocal processing activity between propositional andimplicational levels of meaning. In terms of the further specification of a model of executivefunctions, the error distribution is consistent with the view that evaluations in such novel one-off task environments are not routinely or reliably based upon the propositional content of theoriginal question posed. The pattern observed is consistent with content being primarilyderived and evaluated against an inferred schematic model of number encoded in genericform. Within this approach, the dynamic control of processing activity is inherent within the


representations created and within the processes that act to change one type of mental repre-sentation into another.

Reciprocal processing exchanges between two semantic levels of representation areassumed to underpin performance across a broad range of cognitive tasks, apart from the num-ber generation described here. This obviates the need to postulate generalized, special pur-pose, central executive resources for directing attention, planning, or response selection. Italso enables performance in classic central executive tasks to be related to performance in othertasks normally subject to different types of theoretical treatment and, in doing so, potentiallyfurnish them with a common basis of explanation (see also Barnard, 1999; Teasdale &Barnard, 1993).

REFERENCESAshcraft, M.H., & Battaglia, J. (1978). Cognitive arithmetic: Evidence for retrieval and decision processes in mental

addition. Journal of Experimental Psychology: Human Learning and Memory, 4, 527–538.Baddeley, A.D. (1966). The capacity for generating information by randomisation. Quarterly Journal of Experimental

Psychology, 18, 119–129.Baddeley, A.D., & Hitch, G.J. (1974). Working memory. In G. Bower (Ed.), The psychology of learning and motivation:

Advances in research and theory (pp. 47–90). New York: Academic Press.Barnard, P.J. (1985). Interacting Cognitive Subsystems: A psycholinguistic approach to short-term memory. In A.

Ellis (Ed.), Progress in the psychology of language (Vol. 2, pp. 197–258). London: Lawrence Erlbaum AssociatesLtd.

Barnard, P.J. (1999). Interacting Cognitive Subsystems: Modelling working memory phenomena within a multi-pro-cessor architecture. In A. Miyake & P. Shah (Eds.), Models of working memory. Mechanisms of active maintenanceand executive control. New York: Cambridge University Press.

Barnard, P.J., & Teasdale, J.D. (1991). Interacting Cognitive Subsystems: A systematic approach to cognitive-affec-tive interaction and change. Cognition and Emotion, 5, 1–35.

Benford, F. (1938). The law of anomalous numbers. Proceedings of the American Philosophical Society, 78, 551–572.Dietz, P.A. (1933). Over onderbewuste voorkeur. Nederlandsche tijdschrift voor psychologie, 1, 145–162.Erikson, T.D., & Mattson, M.E. (1981). From words to meaning: A semantic illusion. Journal of Verbal Learning and

Verbal Behaviour, 20, 540–551.Gardner, M. (1973). An astounding self-test of clairvoyance by Dr. Matrix. Scientific American, 229, 98–101.Groen, G.J., & Parkman, J.M. (1972). A chronometric analysis of simple addition. Psychological Review, 79, 329–343.Halliday, M. (1970). Language structure and language function. In J. Lyons (Ed.), New horizons in linguistics (pp.

140–165). Harmondsworth: Penguin Books.Heywood, S. (1972). The popular number seven or number preference. Perceptual and Motor Skills, 34, 357–358.Hockett, C.F. (1967). Where the tongue slips, there slip I. To honour Roman Jakobson, Vol. 2, Janua Lainguarum,

Series Major, Vol. 32.Hsue, E.H. (1948). An experimental demonstration of factor analysis. Journal of General Psychology, 38, 235–241.Kubovy, M. (1977). Response availability and the apparent spontaneity of numerical choices. Journal of Experimental

Psychology: Human Perception and Performance, 3, 359–364.Kubovy, M., & Psotka, J. (1976). The predominance of seven and the apparent spontaneity of numerical choices.

Journal of Experimental Psychology: Human Perception and Performance, 2, 291–294.McCloskey, M., Sokol, S.M., & Goodman, R.A. (1986). Cognitive processes in verbal-number production: Infer-

ences from the performance of brain damaged subjects. Journal of Experimental Psychology: General, 115, 307–330.

Norman, D.A., & Shallice, T. (1986). Attention to action: Willed and automatic control of behaviour. In R.J.Davidson, G.E. Schwarts, & D. Shapiro (Eds.), Consciousness and self regulation, advances in research and theory(Vol. 4, pp. 1–18). New York: Plenum Press.

Poulton, E.C. (1979). Models for biases in judging sensory magnitude. Psychological Bulletin, 86, 777–803.


Power, R.J.D., & Dal Martello, M.F. (1990). The dictation of Italian numerals. Language and Cognitive Processes, 5,237–254.

Power, R.J.D., & Longuet Higgins, H.C. (1978). Learning to count: A computational model of language acquisition.Proceedings of the Royal Society, Series B, 200, 319–417.

Shallice, T. (1988). From neuropsychology to mental structure. Cambridge: Cambridge University Press.Shallice, T., & Evans, M.E. (1978). The involvement of the frontal lobes in cognitive estimation. Cortex, 4, 294–303.Shepard, R.N., Kilpatrick, D.W., & Cunningham, J.P. (1975). The internal representation of numbers. Cognitive

Psychology, 7, 82–138.Teasdale, J., & Barnard, P. (1993). Affect cognition and change: Re-modelling depressive thought. Hove, UK: Lawrence

Erlbaum Associates Ltd.Towse, J.N. (1998). On random generation and the central executive of working memory. British Journal of Psychol-

ogy, 89, 77–101.

Original manuscript received 7 August 1998Accepted revision received 8 May 2000



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APPENDIX BThe frequency of odd and even digits in elaborated responses in Experiments 1, 2, and 3, shown by position in theresponse.

Head of complex Tail of complex Single digitelaboration elaboration elaboration

——————— ——————— ———————i.e., first digit i.e., last digit i.e., single digit

——————— ——————— ———————Condition Odd Even Odd Even Odd Even

Experiment 1 one–ten 16 13 9 6 12 0*nines 32 19 13 19 25 8*7-figure 38 34 25 25 2 1random 26 21 9 17 11 1*

Experiment 2 two–ten 9 6 5 3 5 1nines 23 12 8 11 24 5*7-figure 23 14 8 12 3 0

Experiment 3 one–ten 16 26 11 8 16 1*nines 30 38 16 17 16 5*4-figure 46 33 38 24 – –

Note: The chance occurrence for an odd digit is .55 as zero is not expressed verbally in English numbers. *marks pairsthat are significantly different from chance, p > .05, c2(1) > 3.84.

specifying executive representations and processes in ...specifying executive representations and...

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