Psychological Research *2001) 65: 24±27 Ó Springer-Verlag 2001

ORIGINAL ARTICLE

Michael A. Stadler á David C. Geary á Mary E. Hogan Negative priming from activation of counting and addition knowledge

Received: 15 February 2000 / Accepted: 8 June 2000

Abstract In two experiments, subjects were presented counting are not explicit features of the task *LeFevre, with digit pairs *e.g., 32) and asked to respond to the Bisanz, & Mrkonjic, 1988; LeFevre & Kulak, 1994; rightmost number. Negative priming, that is, slowed Zbrodo€ & Logan, 1986). LeFevre and her colleagues, processing, was evident when the rightmost number was for instance, reported that the presentation of digit pairs, a counting-string *e.g., 43 following 12) or addition-sum such as 34, oftenresulted inthe obligatory activationof *e.g., 65 following 32) associate of the number pair from the associated sum, 7 *LeFevre et al., 1988). Young the preceding trial. The studies are the ®rst to demon- children often retrieve familiar counting-string associ- strate negative priming with counting and arithmetical ates, such as retrieving 3 when presented with 1, 2, in memory representations and suggest the obligatory sequence, even when the task does not involve counting activation of these representations with the presentation *Siegler & Shrager, 1984). of number pairs. The results are also consistent with the If counting-string and addition knowledge do in fact view that negative priming often occurs at the semantic show obligatory activationwhenrelated numbers are level. presented, then negative-priming e€ects should be evident when the activated information is to be ignored on one trial and processed on the next *Driver & Tipper, Introduction 1989; Fox, 1995; May, Kane, & Hasher, 1995; Stadler & Hogan, 1996). In the current studies, subjects were pre- The fast and accurate solving of simple addition prob- sented with pairs of digits, such as 32, and were asked to lems *e.g., 4+5 ˆ ?) and skill at reciting number series press a key corresponding to the rightmost digit, 2 in this *e.g., 1, 2, 3, 4) appear to be dependent on the formation case. If the presentation of these pairs results in obliga- of associations between the presented and related num- tory activationof associated numbers, thenreactiontime bers *Ashcraft & Battaglia, 1978; Campbell, 1995; *RT) should be slowed when the next trial involves re- LeFevre & Bisanz, 1986; Siegler & Shrager, 1984). For sponding to this associate. So, responding to the 5 in the instance, problems such as 5 + 4 can be solved by re- pair 15 should be slower whenit is preceded by the pair trieving the answer, 9, from long-term memory, with the 32 thanwhenit is preceded by the pair 62, for example. retrieval process apparently dependent on the formation Although the mechanisms associated with such neg- of an association between the presented problem and the ative priming are not fully understood, one possibility is answer *Siegler, 1986; see also Shrager & Siegler, 1998). that the to-be-ignored information *i.e., information not Similarly, performance on counting series appears to be directly relevant to the immediate goal) is inhibited so in¯uenced by the strength of associations among the that it will not interfere with the on-line processing of digits in often repeated number strings, such that the the goal-related information *Campbell & Clark, 1989; sequence 1, 2, 3 often leads to the retrieval of 4 *LeFevre Fox, 1995; May et al., 1995). Inthis view, when5 is & Bisanz, 1986; Siegler & Shrager, 1984). automatically activated with the presentation of the 32 Moreover, anobligatory activationof these associa- pair, it is inhibited during the processing of the 2 and tions is found under some conditions where addition or apparently for several seconds thereafter *May et al.). If this is true, then negative priming should be found when target digits are counting-string associates *e.g., 3, fol- M. A. Stadler *&) á D. C. Geary á M. E. Hogan lowing the 12 pair) or the sum *e.g., 6 following the 42 Department of Psychology, University of Missouri-Columbia, 210 McAlester Hall, Columbia, MO 65211, USA pair) of the digit-pair of the preceding trial. Tel.: +1-573-8823009; Fax: +1-573-8827710 Analternativetheory that attributes negativepriming e-mail: [email protected] to episodic retrieval appears to make a similar prediction 25 *e.g., Neill, 1997). With this theory, negative priming Results results from retrieval of a preceding trial based on sim- ilarity betweenit andthe currenttrial. The retrieved Data from three subjects were discarded because they episode, however, includes information activating an failed to follow instructions about responding accurately incorrect response, and this response con¯ict leads to and made errors on more than 10% of the trials; another negative priming. Following up on the example above, if subject's data were discarded because of extremely long the target digit is 3 after subjects respond to 12, then the RTs *nearly double the mean of other subjects; inclusion retrieved episode carries information that 2 is the correct of that subject's data also nearly doubled the standard response and that 3 is an incorrect response, in con¯ict deviation). Accuracy for the remaining subjects was high with the response to be made on the current trial, that is, *M ˆ 92%) and did not vary signi®cantly across con- 3. Presumably retrieval of 12 from the previous trial ditions. Mean RTs for correct responses from the re- would also automatically activate 3, which, of course, is maining subjects are presented in Table 1. Nominally, the correct response for the current trial. Nonetheless, both counting-string associates and addition sums pro- retrieval of 12 should still interfere with the response on duced negative priming. When the target on one trial the current trial and thus slow RTs *e.g. Conway & completed a counting string begun on the preceding Engle, 1994). Our studies were not designed to distin- trial, RTs averaged 21 ms slower thanoncontrol trials. guish betweenthese theories but rather to assess more Similarly, whenthe target ononetrial was the sum of general negative-priming e€ects with the processing of the target and ¯anker from the preceding trial, RTs numerical and addition stimuli. averaged 29 ms slower thanoncontroltrials. Although the former e€ect was not signi®cant, F*1, 11) ˆ 1.32, MSE ˆ 4,132, p > 0.10, the latter one was, F*1, 11) ˆ 5.23, MSE ˆ 1,937, p < 0.05; the two e€ects Experiment 1 did not di€er from one another, F*1, 11) < 1. Method

Subjects. The subjects were 16 University of Missouri-Columbia Experiment 2 students who participated as part of the requirements for Intro- ductory Psychology. One problem with the method of randomly generating trials, as was used in Experiment 1, is that many more Apparatus. Stimulus presentation and response recording were addition trials are produced than counting-string trials. controlled by IBM-compatible computers equipped with VGA This had a direct e€ect onthe reliability of the results of monitors. All timing was in milliseconds, and all stimulus presen- Experiment 1; the standard error for the counting-string tation was synchronized with the raster scan on the monitor. The trials was about twice that of the sum trials. Another stimuli were the digits 1 through 6, which were approximately weakness of Experiment 1 was the relatively small 4 mm high and 3 mm wide and were presented in white on a black background. Subjects responded to the digits by pressing the 1±6 number of subjects. Finally, because the trials were keys at the top of the keyboard, respectively. They responded with generated randomly, there was no control over numer- their left ring *1), left middle *2), left index *3), right index *4), right ous other factors *e.g., the frequency with which various middle *5), and right ring *6) ®ngers. digits appeared as targets) that may have varied across control and negative-priming conditions. Therefore, Procedure. Each trial consisted of the presentation of two digits Experiment 2 was designed to provide a more powerful side-by-side in adjacent columns in the center of the screen *e.g., 12). Subjects were instructed to press the key corresponding to the test of the e€ects of interest by increasing sample size right-hand digit as quickly as possible without making errors on and more systematically generating counting-string and more than5% of the trials. After the subject's response,the screen addition-sum trials. was blank for 500 ms, and then the next trial was presented. At the end of a block of trials, subjects were allowed to rest until they were ready to initiate the next block. Method For each subject, 10 blocks of 100 trials each were generated by randomly selecting a target digit and then randomly selecting one The method was the same as in Experiment 1, with the following of the remaining digits to serve as a ¯anker. On average, this exceptions. procedure produced a total of 14 `counting-string' trials per subject where the target was the next digit in a counting string begun by the Table 1 Mean RT in milliseconds *standard deviations in ¯anker and target in the preceding trial *e.g., 45 followed by 36), parentheses) and priming e€ects as a function of the relation and 61 `addition' trials per subject where the target was the sum of betweenprime andprobe trial stimuli * RT reactiontime) the ¯anker and target in the preceding trial *e.g., 42 followed by 36). Trials in which the target was 3 after the preceding ¯anker was Experimental trials Control trials Priming 1 and the preceding target was 2, which would be both counting string trials and addition trials, were rare and were not analyzed. Experiment 1 Out of the remaining trials, an average of 448 were used as con- Counting string 762 *101) 741 *76) )21 trols. For these, we used only those trials where neither the arith- Sum 770 *85) 741 *76) )29 metic relations just discussed were present, nor any of the relations Experiment 2 typically examined in studies of positive and negative priming *e.g., Counting string 716 *83) 680 *75) )36 repetitionof anignoreditem as a target, or repetitionof a target; Sum 732 *82) 713 *82) )19 see Stadler & Hogan, 1996). 26

Subjects. The subjects were 39 students from the same pool as the presented in Table 1. Both counting-string associates previous experiment. and addition sums produced negative priming. RTs to the probe display were 36 ms slower for counting-string Apparatus. Subjects responded to the digits 1±6 with the corre- trials than for counting-string control trials. Similarly, sponding keys on the numeric keyboard, using their right index ®nger to press the 1 and 4 keys, their right middle ®nger for the 2 RTs to the probe display were 19 ms slower for addition- and 5 keys, and their right ring ®nger for the 3 and 6 keys. sum trials thanfor additioncontrols.Both e€ects were signi®cant, F*1, 33) ˆ 35.30, MSE ˆ 1,270, p < 0.01 for Materials. A set of 360 pairs of displays, each representing a trial, counting-string trials, and F*1, 33) ˆ 12.63, MSE ˆ 955, was used. There were 32 counting-string trials on which the target p < 0.01 for addition-sum trials. The negative-priming digit in the second display completed a counting string begun by e€ect for the addition-sum trials was signi®cantly the ¯anker and target in the ®rst display *e.g., 12 followed by 53); each of the four possible counting strings *123, 234, 345, and 456) smaller, however, than that for the counting-string trials, was used equally often. The ¯anker in the probe display was chosen F*1, 33) ˆ 4.94, MSE ˆ 2,104, p < 0.05. randomly from the digits not already used in the prime and probe Reactiontimes onthe control trials di€ered across displays. Another 32 trials were counting-string control trials. the two conditions, F*1, 33) ˆ 5.76, MSE ˆ 1,637, These trials were identical to the counting-string trials, except that p < 0.05; RT onthe sum controltrials was similar to the ¯anker in the prime display was changed to another digit randomly selected with the constraint that it could not form a probe RT onthe ®ller trials * M ˆ 709), F*1, 33) ˆ 1.02, counting string negative-priming trial *e.g., 42, instead of 12, fol- MSE ˆ 570, but RT on the counting string control lowed by 53) or an addition-sum negative prime trial. trials was faster thanonthe ®ller trials, With the 6 possible target digits, there were 15 possible com- F*1, 33) ˆ 45.30, MSE ˆ 653, p < 0.05. Note that binations of ¯ankers and targets that could sum to the target on the next trial *11, 12, 13, 14, 15, 21, 22, 23, 24, 31, 32, 33, 41, 42, and features of the control trials necessarily varied across 51). Each of these combinations was used twice, for a total of 30 conditions *e.g., the frequency with which various digits trials; the probe display inthose pairs consistedof a target equal to appeared as targets), which could a€ect RTs *Ashcraft, the sum of the ¯anker and target in the prime display and a ¯anker 1995). This was, of course, the reason for having con- chosen randomly from the digits not already appearing in the prime or probe display *e.g., 51 followed by 36). A set of 30 addition trols matched to the negative-priming trials. The im- control trials was generated by changing the ¯anker in the prime portant point is that the probe trials were exactly the display to another digit randomly selected with the constraint that same for the control and negative-priming conditions; it could not form an addition-sum negative-priming trial *e.g., 41, only the contents of the preceding trials varied. That rather than 51, followed by 36) or a counting-string negative prime variationproduced some di€erencesinRT across the trial. Finally, 236 pairs of displays were generated randomly to serve prime trials as well. Reactiontime to prime displays on as ®llers. The total set of 360 displays was thenshu‚ed randomly counting string control trials *M ˆ 733) was slower for each subject. Although the entire sequence of displays was not thanon®ller trials * M ˆ 717), F*1, 33) ˆ 6.92, MSE ˆ entirely random *e.g., there were more counting-string trials than 1,349, p < 0.05, as was RT to prime displays onad- would occur by chance, and the frequencies with which each digit appeared as a target ranged from approximately 40, for the digit 1, dition-sum negative-priming trials, F*1, 33) ˆ 8.92, to 53, for the digit 6), the deviation from complete randomness MSE ˆ 1,694, p < 0.01. Reactiontime to the prime should not have been so great as to lead subjects to develop any display for the counting string *M ˆ 712) and addition- expectancies. Indeed, given the relatively close agreement between sum control *M ˆ 717) trials was not reliably di€erent the results of Experiment 1, where the stimulus sequence was generated randomly, and the results of the present Experiment, from that on®ller trials. where more control was imposed on the sequence, these deviations from randomness do not appear to have had any great impact on subjects' performance. Discussion

Procedure. Each trial consisted of two displays, beginning with the The current studies are the ®rst to demonstrate obliga- prime, which appeared for 750 ms. After the prime disappeared, the screen was blank for 500 ms, and then the probe appeared for tory activation of numerical and arithmetical knowledge 750 ms. After the probe disappeared, the screenwas blankfor by means of negative-priming methodologies and sup- 2,000 ms, and then the next trial began. Responses to the prime and port the view that memory representations of numerical probe were recorded up to 1,200 ms after the onset of the display; information are, at least in part, associative based otherwise, the response was counted as an error. Stimuli were ar- *Ashcraft & Battaglia, 1978; Campbell, 1995; LeFevre ranged on the screen as in the previous experiment, and subjects were instructed to respond to them in exactly the same way. The et al., 1988; Shrager & Siegler, 1998; Siegler & Shrager, 360 trials were presented in 6 blocks of 60. 1984). Negative priming methods provide an advantage over other methods used to study obligatory activation of numerical information, in that the priming e€ect Results provides a quantitative estimate of the strength of the associations. On the basis of the size of the priming ef- Data from ®ve subjects were discarded because they fects *19±36 ms) inthese studies, the strength of the failed to follow instructions about responding accurately counting-string associates and addition-sum associates and made errors on more than 10% of the trials. Accu- appear to be similar, although Experiment 2 suggests racy for the remaining subjects was high *M ˆ 95%) and that the counting-string associations may be slightly did not vary signi®cantly across conditions. Mean RTs stronger. However, individual and developmental dif- for correct responses from the remaining subjects are ferences in the strength of these associations are 27 expected in other samples. For instance, a greater ad- vantage for counting-string over addition-sum associa- References tions is expected for children *Siegler & Shrager, 1984) and the strength of the addition-sum associates is ex- Anderson, M. C., & Spellman, B. A. *1995). On the status of in- hibitory mechanisms in cognition: memory retrieval as a model pected to vary with individual di€erences in arithmetic case. Psychological Review, 102, 68±100. skills *Geary & Brown, 1991; Geary & Widaman, 1987, Ashcraft, M. H. *1995). Cognitive psychology and simple arith- 1992; LeFevre & Kulak, 1994). Further studies are, of metic: A review and summary of new directions. Mathematical course, needed to test these predictions. Cognition, 1, 3±34. Ashcraft, M. H., & Battaglia, J. *1978). Cognitive arithmetic: evi- The results are also relevant to issues in the negative- dence for retrieval and decision processes in mental addition. priming literature. In these studies, negative priming was Journal of Experimental Psychology: Human Learning and observed for semantic associates of the prime stimuli Memory, 4, 527±538. and a€ected the processing of the probe digits 500 ms Campbell, J. I. D. *1995). Mechanisms of simple addition and multiplication: a modi®ed network-interference theory and later. These patterns are in keeping with the view that simulation. Mathematical Cognition, 1, 121±164. negative priming often operates at the semantic level and Campbell, J. I. D., & Clark, J. M. *1989). Time course of error the associated e€ects extend beyond the processing of priming in number-fact retrieval: evidence for excitatory and the prime stimulus *e.g., Driver & Tipper, 1989; Fox, inhibitory mechanisms. Journal of Experimental Psychology: 1995; May et al., 1995). Although the results are not Learning, Memory, and Cognition, 15, 920±929. Conway, A. R. A., & Engle, R. W. *1994). Working memory and de®nitive in this regard, they appear to be consistent retrieval: A resource-dependent inhibition model. Journal of with the view that negative priming results from the Experimental Psychology: General, 123, 354±373. inhibition of information that might disrupt the pro- Driver, J., & Tipper, S. P. *1989). On the nonselectivity of ``selec- cessing of goal-related information *Driver & Tipper, tive'' seeing: contrasts between interference and priming in selective attention. Journal of Experimental Psychology: Human 1989; Fox, 1995; May et al., 1995). Inthis view, the Perception and Performance, 15, 304±314. prime stimuli resulted inthe obligatory activationof Fox, E. *1995). Negative priming from ignored distractors in counting-string or addition-sum associates. The acti- visual selection: a review. Psychonomic Bulletin & Review, 2, vated numbers were not relevant to the goal of identi- 145±173. Geary, D. C., & Brown, S. C. *1991). Cognitive addition: strategy fying and responding to the rightmost digit and, thus, choice and speed-of-processing di€erences in gifted, normal, the activation of these associates was inhibited. Once and mathematically disabled children. Developmental Psychol- inhibition occurs it appears to last for several seconds ogy, 27, 398±406. and thus slowed the processing of the target probes *see Geary, D. C., & Widaman, K. F. *1987). Individual di€erences in May et al., 1995). As noted in the introduction, how- cognitive arithmetic. Journal of Experimental Psychology: General, 116, 154±171. ever, our results might also be accounted for by Neill's Geary, D. C., & Widaman, K. F. *1992). On the convergence of episodic theory *Neill, 1997). convergence of componential and psychometric models. Intel- In any case, the studies demonstrate the utility of ligence, 16, 47±80. negative-priming methodologies for the study of nu- LeFevre, J.-A., & Bisanz, J. *1986). A cognitive analysis of number- series problems: sources of individual di€erences in perfor- merical and arithmetical memory representations and mance. Memory & Cognition, 14, 287±298. these same methods might provide a very useful means LeFevre, J.-A., Bisanz, J., & Mrkonjic, L. *1988). Cognitive for studying individual and group di€erences in the arithmetic: evidence for obligatory activation of arithmetic strength of these associations. At the same time, the facts. Memory & Cognition, 16, 45±53. LeFevre, J.-A., & Kulak, A. G. *1994). Individual di€erences in the studies demonstrate the utility of models of numerical obligatory activationof additionfacts. Memory & Cognition, and arithmetical cognition for informing the study of 22, 188±200. basic issues associated with the negative-priming phe- May, C. P., Kane, M. J., Hasher, L. *1995). Determinants of nomenon *see Ashcraft, 1995, for a review of cognitive negative priming. Psychological Bulletin, 118, 35±54. arithmetic). Neill, W. T. *1997). Episodic retrieval in negative priming and repetitionpriming. Journal of Experimental Psychology: Finally, the development of this paradigm may pro- Learning, Memory, and Cognition, 23, 1291±1305. vide means to simultaneously study inhibitory processes Shrager, J., & Siegler, R. S. *1998). SCADS: A model of children's in selective attention and in memory retrieval using strategy choices and strategy discoveries. Psychological Science, identical methodology. Selective attention is called on to 9, 405±410. Siegler, R. S. *1986). Unities across domains in children's strategy select the target and ignore the distracter, whereas choices. InM. Perlmutter *Ed.), Perspectives for intellectual *presumably automatic) retrieval processes are involved development: Minnesota symposia on child psychology *Vol. 19, inthe activationof the arithmetic facts that thenleads to pp. 1±48). Hillsdale, NJ: Erlbaum. negative priming. Anderson and Spellman *1995) re- Siegler, R. S., & Shrager, J. *1984). Strategy choice inadditionand subtraction: how do children know what to do? In C. Sophian viewed numerous similarities in both the nature of the *Ed.), Origins of cognitive skills *pp. 229±293). Hillsdale, NJ: problems to be solved by mechanisms of selective at- Erlbaum. tention and memory retrieval, and in the phenomena Stadler, M. A., & Hogan, M. E. *1996). Varieties of positive and observed in these domains. The present methodology negative priming. Psychonomic Bulletin & Review, 3, 87±90. may o€er a way to further explore the similarities Zbrodo€, N. J., & Logan, G. D. *1986). On the autonomy of mental processes: a case study of arithmetic. Journal of Exper- betweenthese processes. imental Psychology: General, 115, 118±130.