Graphical representation fosters discovery in the 2-4-6 task

In the 2-4-6 rule discovery task, reasoners seek to discover a rule that governs the arrangement of three numbers (or triple). The to-be-discovered rule is “ascending numbers”. Upon being given the triple 2-4-6 as an initial example, however, reasoners tend to formulate algebraically specific hypotheses. Traditionally, this task is conducted primarily from an internal representation of the triples and candidate hypotheses. More recently, substantial representational effects have been demonstrated wherein an external representation of the dimensions of the problem space facilitated successful rule discovery. In the two experiments reported here, an interactive graphical representation was created by concurrently plotting each triple produced by the participants. In Experiment 1, participants who performed the task with this external representation were more likely to discover the rule than were a group of control participants. Experiment 2 replicated the effect but also assessed participants' hypotheses for each triple generated. Results indicated that a graphical representation of the triples fostered the development of hypotheses that were less constrained by the implied algebraic specificity of the initial triple.     

Hypothesis testing in a rule discovery problem: when a focused procedure is effective.

We investigated individuals' ability to use negative evidence in hypothesis testing. We compared performance in two versions of Wason's (1960) rule discovery problem. In the original version, a triple of numbers--(2, 4, 6)--was presented as an example of a rule that the experimenter had in mind (i.e., "increasing numbers"). Participants had to discover the rule by proposing new triples. In the other version, the same triple was presented as a counter-example to the experimenter's rule (i.e., "decreasing numbers"). We predicted that, in both conditions, participants would form hypotheses based on the features of the triple, and test only instances of the hypothesized rule. However, in the counter-example condition, such focused testing would invariably produce negative evidence. As a consequence, participants would be forced to revise their hypotheses. The reported results corroborated our predictions: Participants solved the counter-example version significantly better than the original problem.     

Hypothesis-testing strategies: Why two goals are better than one

An intriguing finding in the hypothesis-testing literature concerns a large increase in the proportion of subjects who discover a rule when they are asked to determine two rules rather than that rule alone. This finding is based on Wason's (1960) “2 4 6” task, in which subjects try to discover a rule (ascending numbers) by generating and testing number triples. They are initially given an example (“2, 4, 6”) of the rule that leads to overly specific hypotheses. With single-goal (SG) instructions, subjects try to discover the correct rule and are told whether each triple proposed fits the rule. With dual-goal (DG) instructions, correct and incorrect categories are labelled, respectively, as DAX and MED. Subjects try to discover both rules and are told whether each proposed triple is DAX or MED. Two explanations of why DG subjects do better at rule discovery than SG subjects are tested: the quantity of information and the testing of complementary rules using the prevalent positive-test strategy. Results support the latter explanation: DG subjects outperform SG subjects only if they know the rules are complementary, and that SG subjects' performance does not improve when required to test more triples before announcing their first rule. A third explanation, the positivity of the linguistic label of the feedback, is ruled out. Understanding the superiority of DG instructions might suggest a general method for enhancing rule discovery.     

Utilities in the 2-4-6 task

The Wason 2-4-6 task was embedded in a practical reasoning scenario where number sequences had well-defined utilities in the process of achieving a goal. Reasoners' hypothesis-testing behavior was clearly goal-driven and was significantly influenced by whether the utilities favored positive or negative sequences. In the version of the scenario where generating positive sequences had greater benefits than generating negative ones, participants performed poorly at the task as measured by their ability to guess the correct rule and by the nature and number of triples tested before making an announcement. In contrast, the scenario that assigned a greater utility to the production of negative sequences fostered significantly more diligent and creative hypothesis-testing behavior, and participants were significantly more likely to discover the rule. These results suggest that the poor performance observed in Wason's traditional 2-4-6 task reflects a hypothesis-testing process that by default assigns greater utility to the production of sequences that conform to the initial triple, and hence receive positive feedback. However, reasoners are not averse to producing negative sequences, and understand their implication, if their utility is made relevant in the process of achieving goals.     

Inducing a Rule in Wason's 2-4-6 Task: A Test of the Information-Quantity and Goal-Complementarity Hypotheses

In the standard 2-4-6 induction task, subjects are instructed to discover the rule generating sequences of three numbers by inventing number triples for which they receive immediate feedback. The rule is “ascending numbers”. Performance is greatly aided with Dual Goal (DG) instructions that ask subjects to discover two rules, one that generates “Dax” triples (equivalent to “yes” instances with Single Goal [SG] instructions) and another that generates “Med” triples (equivalent to “no” instances). The present study eliminates two explanations for this effect suggested by Wharton, Cheng, and Wickens (1993). Experiment 1 tested their Information-Quantity hypothesis that the effect results simply from the DG subjects testing more triples prior to proposing a rule. Our DG subjects were more likely to solve the problem and produced more “negative” triples than SG subjects when both groups generated exactly 15 triples. Two further groups received feedback only after generating all 15 triples, and again DG subjects were more likely to solve the problem and to generate more “negative” triples. Experiment 2 tested Wharton et al.'s Goal-Complementarity hypothesis that success under DG instructions hinges on preserving the complementary representation of the two rules. We compared SG instructions with three types of DG instructions that suggested different types of triples (Dax, Med, both Dax and Med, neither Dax nor Med). DG instructions were more effective in promoting successful rule discovery regardless of differences in rule complementarity. Our analysis of the heterogeneity of the examplars generated with DG instructions in both experiments suggest that success on the 2-4-6 task is as much a consequence of the breadth of hypotheses that subjects entertain as it is a consequence of the testing strategy.     

Error, falsification and scientific inference: An experimental investigation

This paper reports the results of four experiments designed to test the methodological falsificationist's assumption that replication is sufficient to prevent the possibility of error from being used to immunize hypotheses against disconfirmation. The first three experiments compare the performance of subjects on tasks that simulate scientific reasoning under two conditions: (1) where there is a 0-20% possibility of error in experimental results, but no actual error; and (2) a control condition.

All experiments used Wason's 2-4-6 task, in which subjects propose triples and are told whether each corresponds to a rule. In Experiment 1, subjects in the possible-error condition proposed significantly more triples than control subjects. Experiment 2 added colour and letter dimensions to the 2-4-6 task; possible-error subjects proposed significantly more triples and replicated the same triple more often than control subjects. Experiment 3 made replication more difficult by limiting the number of experiments subjects could perform and by altering the rule to make the results of the current trial dependent on previous ones. Control subjects solved this problem significantly more often than possible-error subjects.

Experiment 4 was run in a manner very similar to Experiment 1, except that an actual 20% error condition was added. Subjects in this condition solved the rule significantly less often than subjects in other conditions, and also took more time and replicated more often. Implications of these results for the methodological falsificationist's position are discussed.     

Creativity and inductive reasoning: The relationship between divergent thinking and performance on wason's 2-4-6 task

This study was an investigation of the relationship between potential creativity—as measured by fluency scores on the Alternate Uses Test—and performance on Wason's 2-4-6 task. As hypothesized, participants who were successful in discovering the rule had significantly higher fluency scores. Successful participants also generated higher frequencies of confirmatory and disconfirmatory hypotheses, but a multiple regression analysis using the stepwise method revealed that the frequency of generating disconfirmatory hypotheses and fluency scores were the only two significant factors in task outcome. The results also supported earlier studies where disconfirmation was shown to play a more important role in the later stages of hypothesis testing. This was especially true of successful participants, who employed a higher frequency of disconfirmatory hypotheses after receiving feedback on the first announcement. These results imply that successful participants benefited from the provision of feedback on the first announcement by switching to a more successful strategy in the hypothesis-testing sequence.     

A philosophical and empirical analysis of subjects' modes of inquiry in wason's 2-4-6 task

Investigations of hypothesis-testing behaviour typically conclude that subjects' methods are characterized by “confirmation bias.” However, these studies (a) relied exclusively upon Karl Popper's philosophy of science, (b) only described subjects' strategies via a falsificationist terminology, and (c) failed to determine subjects' precise intentions in the task situations. Alternative philosophies of science can be used to describe hitherto ignored aspects of rule discovery in Wason's (1960) 2-4-6 task as well as reinterpret findings from previous research. Using Wason's task, Experiment 1 adopted J. S. Mill's inductive methodology in order to assess better the extent of elimination in experimentation. While it was found that subjects did eliminate a large proportion of their hypotheses rather than merely confirm them, Mill's eliminative methods applied to this task failed to account for the entirety of subjects' behaviours. Experiment 2 examined subjects' methods in more depth by asking subjects during the task either to: (1) describe their procedures, (2) select from a list of methodology terms those that best characterized their methods, and (3) state on each learning trial their preferred hypotheses and their subjective assessments that those hypotheses were correct. Although subjects were generally unable to describe their methods of inquiry, they were able to typify them via the methodology terms as if their methods were tacitly held skills. Contrary to presumptions of previous research, analyses indicated that subjects were not testing particular hypotheses on every trial but were often examining possible instances of the rule “at random” or “different from ones examined” to that trial. It is also suggested that subjects' decisions to announce rules and their thematic lines of inquiry be given further consideration by researchers.     

Transfer of conditional reasoning: Effects explanations and initial problem types

Transfer of reasoning on Wason's (1966) selection task was explored in three experiments. Experiment 1 tested the effects of problem explanations and verbalization instructions on transfer from abstract or thematic problems to abstract problems. Explanations facilitated transfer only when the initial problems were abstract; verbalization did not produce transfer between problems. Experiment 2 explored the effects of problem similarity and explanations on transfer between problems. Although transfer occurred following explanations, no effect of similarity was found for thematic problems. In both of these experiments, the thematic effect (Wason & Shapiro, 1971) was observed. Experiment 3 examined the effects of explanations to abstract or thematic problems on transfer to subsequent abstract or thematic problems. Transfer of reasoning occurred from both initial problem types, particularly to problems of the same type; however, transfer occurred to a greater extent from abstract problems than from thematic problems. The results are discussed in terms of problem similarity and Cheng and Holyoak's (1985) pragmatic reasoning schema hypothesis.     

Focusing on the relation: fewer exemplars facilitate children's initial verb learning and extension

One of the most prominent theories for why children struggle to learn verbs is that verb learning requires the abstraction of relations between an object and its action (Gentner, 2003). Two hypotheses suggest how children extract relations to extend a novel verb: (1) seeing many different exemplars allows children to detect the invariant relation between actions in different contexts (Gentner, 2003), and (2) repetition of fewer exemplars allows children to move beyond the entities involved to extract the relation (Kersten & Smith, 2002). We tested 2 1/2- and 3-year-olds' ability to extend a novel verb after viewing the repetition of one novel actor compared to four different actors performing a novel action. Both ages were better at learning and extending a novel verb to a novel actor when shown only one actor rather than four different actors. These results indicate that during initial verb learning less information is more effective.     

Disconfirmation and dual hypotheses on a more difficult version of wason's 2-4-6 task

This paper reports two experiments in which subjects worked to solve a more difficult version of Wason's 2-4-6 task: instead of the usual “numbers must ascend in order of magnitude” rule, a more general rule, “the three numbers must be different”, was used. The first experiment established that instructing subjects to disconfirm on the “three different numbers” did not significantly improve their performance, as compared with confirmatory and control groups. Disconfirmatory subjects did try to propose more triples at variance with their hypotheses but were unable to obtain the necessary disconfirmatory information.

To help subjects represent the task in a way that facilitated disconfirmation, the second experiment utilized a procedure in which subjects were told that they were looking for two rules, Dax and Med-the Dax rule corresponding to “three different numbers” and the Med rule to its complement, i.e. two or more numbers the same. Of subjects in the Dax-Med condition, 88% solve the rule, as opposed to 21% of subjects in a control condition. Dax-Med subjects tended to search for positive instances of the Med rule, which, in turn, forced them to test the limits of the Dax rule. It was concluded that the Dax-Med manipulation did facilitate a different mental representation of the task than the normal procedure.     

Social cues modulate the representations underlying cross-situational learning

Because children hear language in environments that contain many things to talk about, learning the meaning of even the simplest word requires making inferences under uncertainty. A cross-situational statistical learner can aggregate across naming events to form stable word-referent mappings, but this approach neglects an important source of information that can reduce referential uncertainty: social cues from speakers (e.g., eye gaze). In four large-scale experiments with adults, we tested the effects of varying referential uncertainty in cross-situational word learning using social cues. Social cues shifted learners away from tracking multiple hypotheses and towards storing only a single hypothesis (Experiments 1 and 2). In addition, learners were sensitive to graded changes in the strength of a social cue, and when it became less reliable, they were more likely to store multiple hypotheses (Experiment 3). Finally, learners stored fewer word-referent mappings in the presence of a social cue even when given the opportunity to visually inspect the objects for the same amount of time (Experiment 4). Taken together, our data suggest that the representations underlying cross-situational word learning of concrete object labels are quite flexible: In conditions of greater uncertainty, learners store a broader range of information.     

Evaluating negative tests and refutations in a rule discovery task

Subjects participating in Wason's rule discovery task (1960) overwhelmingly try to confirm rather than refute their currently held hypothesis. Such a strategy is often inadequate and runs counter to the canons of scientific methodology. The present study was designed to investigate subjects' differential evaluation of test strategies and outcome. One-hundred and sixty students participated in two experiments in which they had to judge someone else's potential test items in Wason's task. Experiment 1 demonstrated that exposure to various histories has a mediating effect on the strength of the confirmation bias. In Experiment 2, subjects knew the researcher's rule and thus whether each proposed test item would lead to confirmation or refutation of the hypothesis under consideration. The preferred items were those that alerted the subject to an incorrect hypothesis (refutation) and those that turned out to be positive instances of the rule sought after, with the combination of the two (a negative test leading to refutation) being most highly evaluated.     

Priming 4.5-month-old infants to use height information by enhancing retrieval

How do infants select and use information that is relevant to the task at hand? Infants treat events that involve different spatial relations as distinct, and their selection and use of object information depends on the type of event they encounter. For example, 4.5-month-olds consider information about object height in occlusion events, but infants typically fail to do so in containment events until they reach the age of 7.5 months. However, after seeing a prime involving occlusion, 4.5-month-olds became sensitive to height information in a containment event (Experiment 1). The enhancement lasted over a brief delay (Experiment 2) and persisted even longer when infants were shown an additional occlusion prime but not an object prime (Experiment 3). Together, these findings reveal remarkable flexibility in visual representations of young infants and show that their use of information can be facilitated not by strengthening object representations per se but by strengthening their tendency to retrieve available information in the representations.     

Mental representation and hypothetico-deductive reasoning: The case of the THOG problem

Summary This article reports three experiments that deal with the source of the difficulty of Wason's (1977) THOG problem. The solution of this problem demands both the postulation of hypotheses and a combinatorial analysis of their consequences. Experiment 1 showed that the generation of the hypotheses is not in itself sufficient to solve the problem. Experiment 2 showed that a version presenting a plausible context for separating the level of data from that of hypotheses produced a better performance than both the original abstract version and a thematic version lacking the plausible context separating the levels. Experiment 3 gave evidence that this context can produce facilitation even with the geometric material of the classic version. This experiment also showed that a pictorial presentation of data and a verbal presentation of hypotheses affect performance negatively. The results demonstrate the role of problem representation in problem solving, and, in particular, the role of homogeneity in representing data and hypotheses in hypothetico-deductive reasoning.     

Statistical regularities guide the spatial scale of attention

Visual scenes contain information on both a local scale (e.g., a tree) and a global scale (e.g., a forest). The question of whether the visual system prioritizes local or global elements has been debated for over a century. Given that visual scenes often contain distinct individual objects, here we examine how regularities between individual objects prioritize local or global processing. Participants viewed Navon-like figures consisting of small local objects making up a global object, and were asked to identify either the shape of the local objects or the shape of the global object, as fast and accurately as possible. Unbeknown to the participants, local regularities (i.e., triplets) or global regularities (i.e., quadruples) were embedded among the objects. We found that the identification of the local shape was faster when individual objects reliably co-occurred immediately next to each other as triplets (local regularities, Experiment 1). This result suggested that local regularities draw attention to the local scale. Moreover, the identification of the global shape was faster when objects co-occurred at the global scale as quadruples (global regularities, Experiment 2). This result suggested that global regularities draw attention to the global scale. No participant was explicitly aware of the regularities in the experiments. The results suggest that statistical regularities can determine whether attention is directed to the individual objects or to the entire scene. The findings provide evidence that regularities guide the spatial scale of attention in the absence of explicit awareness.     

Adjectives really do modify nouns: the incremental and restricted nature of early adjective acquisition

By 24 months, most children spontaneously and correctly use adjectives. Yet prior laboratory research that has studied lexical acquisition in young children reports that children up to 3-years-old map novel adjectives to object properties only in very limited situations (Child Development 59 (1988) 411; Child Development 64 (1993) 1651; Child Development 71 (2000) 649; Developmental Psychology 36 (2000) 571; Child Development 69 (1998) 1313). In Experiments 1 and 2 we introduced 36-month-olds (Experiment 1) and 24-month-olds (Experiment 2) to novel adjectives while providing rich referential and syntactic information to indicate what the novel words mean. Specifically, we used a given novel adjective to describe multiple familiar objects which shared a salient property; in addition we used the adjectives in full noun phrases, not in conjunction with pronouns. Under these conditions, both groups mapped novel adjectives onto object properties. In Experiment 3 we asked whether the rich referential information was responsible for the successful outcome of the previous two experiments; we introduced novel adjectives to 2- and 3-year-olds as in Experiments 1 and 2, but the adjectives modified nouns of vague (very general) reference ("one", or "thing"). Under these conditions the children failed. We suggest that young word learners require access to the taxonomy of the object type so that the relevant property can be identified. The taxonomically specific nouns of Experiments 1 and 2 accomplish this, whereas the more general, semantically bleached nominals in Experiment 3 do not. Taken together with related findings in the literature, these findings favor an account of lexical acquisition in which layers of information become available incrementally, as a consequence of solving prior parts of the learning problem.     

Speech segmentation by statistical learning depends on attention

We addressed the hypothesis that word segmentation based on statistical regularities occurs without the need of attention. Participants were presented with a stream of artificial speech in which the only cue to extract the words was the presence of statistical regularities between syllables. Half of the participants were asked to passively listen to the speech stream, while the other half were asked to perform a concurrent task. In Experiment 1, the concurrent task was performed on a separate auditory stream (noises), in Experiment 2 it was performed on a visual stream (pictures), and in Experiment 3 it was performed on pitch changes in the speech stream itself. Invariably, passive listening to the speech stream led to successful word extraction (as measured by a recognition test presented after the exposure phase), whereas diverted attention led to a dramatic impairment in word segmentation performance. These findings demonstrate that when attentional resources are depleted, word segmentation based on statistical regularities is seriously compromised.     

Learning at a distance I. Statistical learning of non-adjacent dependencies

In earlier work we have shown that adults, young children, and infants are capable of computing transitional probabilities among adjacent syllables in rapidly presented streams of speech, and of using these statistics to group adjacent syllables into word-like units. In the present experiments we ask whether adult learners are also capable of such computations when the only available patterns occur in non-adjacent elements. In the first experiment, we present streams of speech in which precisely the same kinds of syllable regularities occur as in our previous studies, except that the patterned relations among syllables occur between non-adjacent syllables (with an intervening syllable that is unrelated). Under these circumstances we do not obtain our previous results: learners are quite poor at acquiring regular relations among non-adjacent syllables, even when the patterns are objectively quite simple. In subsequent experiments we show that learners are, in contrast, quite capable of acquiring patterned relations among non-adjacent segments-both non-adjacent consonants (with an intervening vocalic segment that is unrelated) and non-adjacent vowels (with an intervening consonantal segment that is unrelated). Finally, we discuss why human learners display these strong differences in learning differing types of non-adjacent regularities, and we conclude by suggesting that these contrasts in learnability may account for why human languages display non-adjacent regularities of one type much more widely than non-adjacent regularities of the other type.     

Cue familiarity but not target retrievability enhances feeling-of-knowing judgments.

Two hypotheses concerning people's ability to predict later memory performance for unrecalled items were investigated. The target retrievability hypothesis states that feeling-of-knowing judgments (FKJs) are based on partial target information; and the cue familiarity hypothesis asserts that they are based on recognition of the cues. In Experiments 1 and 2, subjects either generated or read the targets of paired associates. Half of the cues had been primed in a pleasantness-rating task. The generation manipulation increased recall but had no effect on FKJs. Cue priming had no effect on recall but increased FKJs. In Experiment 3, using general information questions, primed after the initial recall attempt, both cue and target priming increased FKJs. Experiment 4, which remedied difficulties in Experiment 3, showed no effect of target priming whereas cue priming increased FKJs. The results favor the cue familiarity hypothesis.