Simple arithmetic processing: Surface form effects in a priming task

Abstract

Models of numerical processing vary on whether they assume common or separate processing pathways for problems represented in different surface forms. The present study employed a priming procedure, with target naming task, in an investigation of surface form effects in simple addition and multiplication operations. Participants were presented with Arabic digit and number word problems in one of three prime-target relationships, including congruent (e.g., ‘2 + 3’ and ‘5’), incongruent (e.g., ‘9 + 7’ and ‘5’) and neutral (e.g., ‘X + Y’ and ‘5’) conditions. The results revealed significant facilitatory effects in response to congruent digit stimuli at SOAs of 300 and 1000 ms, in both operations. In contrast, inhibitory effects were observed in response to incongruent word stimuli in both the addition and multiplication operations at 300 ms, and in the addition operation at 1000 ms. The overall priming effects observed in the digit condition were significantly greater than in the word condition at 1000 ms in the multiplication operation and at 300 ms in the addition operation. The results provide support to separate pathway accounts of simple arithmetic processing for problems represented in different surface forms. An explanation for variation in processing due to differences in access to visual and phonological representations is provided.

Introduction

Do the surface characteristics of arithmetic problems (e.g., Arabic digits: 2 + 3; written number words: two + three) influence cognitive processing? This question is central to much of the research undertaken in the past three decades in the cognitive arithmetic area, having implications for models describing the componential architecture of numerical knowledge and the access to this information in the brain (Ashcraft, 1992, Campbell, 1994, Campbell, 1999, Dehaene, 1992, Noel et al., 1997). Four main models of numerical processing are prominent in the literature, including the abstract-modular model (McCloskey, Caramazza, & Basili, 1985), the triple code model (Dehaene, 1992), the preferred entry code model (Noel & Seron, 1993), and the encoding complex hypothesis (Campbell & Clark, 1988; see Noel et al. (1997) for a review of these models). Importantly, all of the numerical processing models assume that problems represented in different surface forms can be converted to the same mental representation and then processed along a common pathway. However, the encoding complex hypothesis differs from the other models in that it also assumes that problems represented in different surface forms can remain different and can be individually processed along separate pathways i.e., as specific codes (Campbell & Clark, 1988).

Empirical support for the notion that separate pathways can be used to process numbers represented in different surface forms is provided in a series of investigations into simple arithmetic fact retrieval. In the first of these, Campbell and Clark (1992) tested one of the main assumptions underlying McCloskey et al.’s (1985) abstract modular model, which suggests that fact retrieval is achieved through the operation of an independent calculation module and therefore, is a process that is not sensitive to the initial form of a problem. The participants in this study were asked to retrieve solutions to simple multiplication problems represented as either Arabic digits or written number words. The results revealed an interaction between problem size and surface form, with a greater increase in reaction times and error rates for larger problems following presentation of the word stimuli. Furthermore, a regression analysis showed that variables that were theoretically related to retrieval difficulty and interference (i.e., problem size – where reaction time and errors increase with problem magnitude, and fan – problems that share solutions produce greater reaction times) predicted word-digit differences. These findings were considered not easily reconcilable with the abstract modular model’s assumption that number fact retrieval is mediated by a single, format independent, abstract representation.

In response to this, McCloskey, Macaruso, and Whetstone (1992) argued that the digit and word form differences identified by Campbell and Clark (1992) were possibly the result of encoding differences, with fact retrieval for word problems being carried out under greater speed pressure than for digit problems. According to McCloskey et al. (1992), this occurred for two main reasons. Firstly, the encoding of words requires the processing of several characters spread over a greater physical length than digits, thereby necessitating longer encoding times for word problems. Secondly, substantial frequency differences occur not only between the words two and nine but also between words and digits, a factor that Campbell and Clark (1992) had failed to consider. In support of their argument, McCloskey et al. (1992) repeated Campbell and Clark’s (1992) regression analysis, with predictor variables that included the number of characters comprising each problem and frequency, and found that the problem size and fan effects disappeared. Furthermore, in view of these encoding effects, McCloskey et al. (1992) argued that if participants were to adopt a response deadline that limited the amount of time between exposure to the problem and responding, word problems would be subject to less processing in the retrieval stage, potentially increasing error rates for larger problems and the incidence of numerically distant errors. Nevertheless, in their study, Campbell and Clark (1992) concluded that the surface form effects had ‘emerged over and above encoding effects’ and further supported their claim with an in depth analysis of errors in performance that suggested an interaction between number-reading processes and number fact retrieval (p. 478; but see Noel et al., 1997, for a critical review of the interpretation of error data). As noted by Campbell (1994), such a finding was inconsistent with the abstract-modular model, which holds that these two processes should not interact.

In a subsequent study by Campbell (1994) that also included an addition condition, the problem size and surface form variables were again shown to interact. In addition to this, the results revealed word format costs in reaction time that were greater for the larger, more difficult problems in the addition condition than in the multiplication condition. With the same operands utilised for both operations, the finding of an operation-by-format-by-size interaction was difficult to explain in terms of encoding processes (Campbell, 1994, Noel et al., 1997). However, as Campbell (1994) himself noted, given the possibility that the effects of problem size vary as a function of operation, it is plausible that the processing of attention-demanding larger problems (e.g., 9 + 5 = 10 + 5 − 1) would be interfered with more by the encoding of problems that required greater attentional resources i.e., the encoding of problems represented in a word format.

Following the initial suggestion by McCloskey et al. (1992), that Campbell and Clark’s (1992) findings might be explained in terms of encoding processes and the acknowledgement of this possibility in Campbell’s (1994) study, a number of studies were undertaken that attempted to separate the effects of encoding from fact retrieval processes. In one such study, Noel et al. (1997) reasoned that if the interaction obtained in the multiplication task was due mainly to encoding processes, then a similar interaction should be found in a non-arithmetic task that involved similar encoding processes. Participants in this study were first asked to produce the solutions to multiplication problems represented in digit and word formats and then to perform a number matching task on the same pairs of digits and words. In the latter case, participants were first exposed to two canonical dot patterns and then were presented with either a pair of digits or a pair of number words. Their task was simply to indicate whether the digits or words represented the same numerosities as those expressed by the dots. The results revealed a similar format-by-size interaction in both the fact retrieval and the number matching tasks, thereby supporting an encoding based account of Campbell’s (1994) findings.

However, the possibility exists that the number matching task employed by Noel et al. (1997) may have unintentionally confounded encoding processes with obligatory fact retrieval processes (which are also shown to produce problem size effects, e.g., see Jackson and Coney, 2005, Jackson and Coney, in press). For example, in a study by LeFevre, Bisanz, and Mrkonjic (1988), participants were presented with two numbers (e.g., 3 + 2) and were then required to decide if a target number (e.g., 5) was one of the original numbers presented. Lengthier decision times in responding to the correct sum following the presentation of simple addition problems were found. Moreover, this effect was found even without the presence of the arithmetic operator (e.g., 3 2) showing that the obligatory activation of simple arithmetic facts occurs simply as the result of exposure to a pair of numbers. This finding was later supported in a similar study of the multiplication operation by Thibodeau, LeFevre, and Bisanz (1996), although in this case, the arithmetic operator was included in all conditions. It is at least possible therefore, that the number matching task employed by Noel et al. (1997) may have inadvertently accessed fact retrieval processes, hence producing the same format-by-size interaction as that in their multiplication task.

In another study by Campbell (1999), the influence of encoding in the format-by-size interaction was investigated using simple addition stimuli and the simultaneous or sequential presentation of operands (also see Blankenberger & Vorberg, 1997, who employed a similar methodology). In the simultaneous condition, i.e., the standard method of stimulus presentation, the usual interaction was predicted by Campbell (1999). However, in the sequential condition, the right operand was presented 800 ms after the left operand, thereby allowing time for the left operand to be processed before the right one was presented. Campbell (1999) argued that the encoding differences should therefore arise only in connection with the second operand and if the format-by-size interaction occurred mainly at the encoding stage, its magnitude should be reduced by half when compared to the simultaneous condition. The results showed that the interaction did not differ between simultaneous and sequential conditions leading Campbell to conclude that it did not occur at the encoding stage but instead arose during calculation or production.

Nevertheless, it is questionable as to whether the simplistic interpretation of the encoding process in the sequential condition described by Campbell (1999) is what actually occurs. For example, if access to a correct arithmetic solution requires the encoding of the problem as a whole (e.g., see Blankenberger & Vorberg, 1997, or Campbell, 1987, Campbell and Graham, 1985, Network Interference model of arithmetic processing) then potentially, the encoding process in this condition will be more complex, requiring the integration of the numerical representation of the right operand with the left operand and operator held in short term memory. Then, with both methods of presentation ultimately requiring whole problem encoding, the same format-by-size interaction should be found. Whatever the case may be, the issue is that any assumptions made regarding the encoding and fact retrieval stages associated with each condition, at this point, are speculative at best.

More recently, Campbell and Fugelsang (2001) investigated the format-by-size interaction by exploring the notion that surface form effects could arise from differences in the choice of strategy employed to access arithmetic solutions. According to Campbell and Fugelsang, because simple arithmetic problems are rarely encountered as words, visual familiarity with these problems will be low. This, together with the robust finding of greater problem difficulty with word stimuli, may promote the use of calculation strategies (e.g., counting or transformation: 6 + 7 = 6 + 6 + 1) and discourage the use of direct memory retrieval, which is possibly more likely to be used with the more familiar digit stimuli. To test this hypothesis, a verification procedure that required participants to indicate whether addition problems presented as digits (3 + 4 = 8) or words (three + four = eight) were true or false was employed in conjunction with self report measures of the participants’ solution strategies. The results revealed the same format-by-size interaction in reaction times that was recognised in earlier production and matching tasks. Furthermore, the reported use of calculation strategies was found to be much greater for word stimuli than digit stimuli, a difference that was exaggerated for larger problems. Accordingly, the findings were again interpreted as evidence for surface form effects in central, rather than encoding stages of processing.

However, a recent study by Smith-Chant and LeFevre (2003) showed that in simple arithmetic processing, individual differences in arithmetic fluency and instructional demands can bias self reports and the solution procedures that are described. In this study, participants were asked to solve single digit multiplication problems under both speed and accuracy instructions and then half of the participants provided self reports of their solutions to the problems. Low skilled participants were shown to respond more slowly and accurately when asked to describe their solution procedures for large and very large problems. Moreover, they were more likely to use a greater variety of procedures, altering these with changes in emphasis on instructions between speed and accuracy. Unfortunately, Campbell and Fugelsang (2001) did not consider skill level at the time that they conducted their study.

Thus, regardless of ‘considerable experimental effort’, the question of just what influence encoding processes have in producing the format-by-size interaction remains largely unanswered (Campbell, 1999, p. B26). As noted by McCloskey et al. (1992), unless subjective size differences between large and small stimuli are made equivalent for each format, size incongruity effects cannot meaningfully be compared between formats. Possibly as a consequence of this, in the final example of a study that addressed the issue of surface form in numerical processing and that attempted to isolate the effects of encoding from fact retrieval processes, the influence of problem size in processing was not considered.

In Experiment 1 of a repetition priming investigation, Sciama, Semenza, and Butterworth (1999) presented participants with addition problems represented as Arabic digits and number words. In Experiment 2, the addition problems were represented as Arabic digits and dot configurations. In each experiment, one third of the problems were preexposed in the same notation, one third were preexposed in a different notation, and one third were not preexposed. Participants were simply asked to sum the numbers. The results indicated that preexposure to the same number pair represented in the same form produced greater benefits in reaction time for word and dot stimuli than did preexposure of the same number pair in digit form. With addition problems seldom ever represented using number words or dots, the authors concluded that the influence of surface form on repetition priming was dependent on the typicality of the surface form for that task. However, in addition to this, the results also revealed priming effects across surface form. That is, preexposure to the same number pair represented as digits, words or dots led to the same amount of priming in digit stimuli. Such a finding is consistent with models that assume that after encoding, processing involves a common representation. The results of the Sciama et al. (1999) study therefore, supported the encoding complex hypothesis and the notion that both common and form specific codes co-exist together.

Nonetheless, as noted by Sciama et al. (1999), it is possible that the surface form effects observed for the word and dot stimuli in their first two experiments resulted from facilitated encoding processes, due simply to exposure to atypical stimuli. Consequently, in Experiment 3 of their study, the authors reasoned that if this was the case, priming should be found for the same numbers presented in different operations (e.g., 2 + 3 and 2 × 3) for the word and dot stimuli alone. To test this, the same method as that employed in the first two experiments was utilised but this time, the surface form was maintained across repetitions. Additionally, three study phases were employed, the first of which, required participants to perform multiplication on the prime instead of addition. Of the remaining study trials, one third of the items were not presented at study (i.e., they were new in the test phase) and the other third were presented for addition. The results suggested priming for number pairs that had been multiplied in the study phase, and priming reached significance when the number pairs had to be added at study. Furthermore, this trend for cross operation priming was apparent for all surface forms, and was more reliable with the digit stimuli. The findings were thus deemed inconsistent with models that explain effects of surface form in terms of encoding processes.

In the cognitive arithmetic literature, models of numerical processing differ on the fundamental issue of whether the surface characteristics of arithmetic problems influence later cognitive processing. That is, there is disagreement as to whether problems represented in different surface forms are first converted to a single representation before processing along a common pathway or remain unique, and are processed individually as specific codes. Underlying this disagreement, there appears to be an inability to reliably determine whether the surface form effects (e.g., the format-by-size interaction) that are robustly identified in simple arithmetic tasks result due to encoding or fact retrieval mechanisms.

The aim of the present study was thus to resolve this problem by utilising an arithmetic based variant of the single word semantic priming paradigm in the investigation of multiplication and addition processing (e.g., see Jackson and Coney, 2005, Jackson and Coney, in press). This priming procedure differed from earlier cognitive arithmetic priming investigations (e.g., see Campbell, 1987, Campbell, 1991) in that it involved the presentation of problems as primes (e.g., 2 + 3) and solutions as targets (e.g., 5), in the order that they occur in natural settings. Moreover, the time period between the onset of the prime and presentation of the target (i.e., the stimulus onset asynchrony; SOA) was varied in order to assess automatic and strategic processing. In line with the single word semantic priming paradigm in which automatic effects are measured at SOAs in the order of 250 ms and strategic effects are measured at SOAs of greater than 400 ms, the present study employed SOAs of 300 and 1000 ms (Perea and Rosa, 2002, Velmans, 1999). When used in conjunction with a target naming (i.e., pronounciation) task, this procedure allowed for a more valid investigation into automaticity in arithmetic fact retrieval than that which occurs with verification or production tasks. This is because, in both verification and production tasks, faster responses and greater accuracy are attributed to automatic processing. However, there is little basis for determining where the boundary is in the range of reaction time and error rate measures that separate the operation of automatic and strategic fact retrieval mechanisms. Furthermore, verification tasks may induce attentional processing through the requirement to make a binary decision about the relationship between the prime and the target, and may be accomplished via processes other than fact retrieval, including familiarity, plausibility and odd/even judgements (Campbell, 1987). Thus, by simply requiring that participants’ verbally identify target numbers as they appeared on a computer screen, the naming task minimised the possibility of calculation and decision induced attentional processing.

Importantly, in the context of the present study, the use of this priming procedure allowed for a comparison of the priming effects produced by exposure to each surface form (i.e., rather than making direct comparisons of reaction times between digits and words). To do this, simple addition and multiplication problems represented in each surface form were assigned to three prime-target relationship conditions i.e., congruent (‘2 + 3’ and ‘5’), incongruent (‘7 + 9’ and ‘5’) and neutral (‘X + Y’ and ‘5’) conditions. Consistent with Neely (1991), the effects of the congruent and incongruent prime-target relationships were then assessed independently for each surface form by subtracting the reaction time taken to name the targets in each of these conditions from the reaction time taken to name the target following exposure to the neutral condition. Positive differences were referred to as facilitation and negative differences were referred to as inhibition. Additionally, by subtracting the reaction time taken to name the targets in the congruent condition (e.g., ‘2 + 3’ and ‘5’) from the reaction time taken to name the targets in the incongruent condition, in which the same prime was presented (i.e., ‘2 + 3’ and ‘14’), an overall priming effect that was independent of encoding times was produced for each surface form. Accordingly, it was assumed that if problems represented as digits and words are accessed via common pathways, then the patterns of priming effects that they each produce would not differ.