A central question in mathematical cognition concerns the nature of the processes involved in mental arithmetic. Over the years, several competing models of mental arithmetic have been proposed. While most models share a serial architecture of encoding, calculation (which may include retrieval), and production (Ashcraft, 1992), these competing models differ with respect to the proposed independence of these stages. The abstract code model (McCloskey, 1992; McCloskey et al., 1992; McCloskey and Macaruso, 1995) proposes separate encoding, calculation, and production modules that communicate via an abstract semantic representation. Each module is specialized for a particular type of input; that is, there are separate encoding modules for verbal numerals (e.g., “four”) and Arabic numerals (e.g., “4”). For example, consider the problem 3 + 4. According to the abstract code model, this problem would be solved by first entering through a comprehension module, specialized for the type of stimulus (in this case, Arabic numerals). After this initial encoding, the problem would then be converted to an amodal, semantic representation (an “abstract code”). Calculation (e.g., retrieval) would operate on this abstract code. The result of the calculation (the answer 7, still in the form of an abstract representation) would then feed into a production module, specialized for the type of production required in the task (either verbal or Arabic).

On the other hand, the triple code model (Dehaene, 1992; Dehaene and Cohen, 1995) proposes three separate modules, each specialized for a specific type of representational format: an analog magnitude representation (e.g., a mental number line), an auditory/verbal module, and a visual Arabic numeral module. The triple code model differs from the abstract code model in that calculation and production occur within each module. For comparison, consider again our example 3 + 4. In the triple code model, this problem would be input into one of two modules, each specialized for the type of input code (either an auditory-verbal word frame or a visual-Arabic number form). Calculation would then take place within one of these modules, depending on the nature of the problem. As an example, a retrieval-based calculation to a visually-presented problem (e.g., “3 + 4”) could proceed by first transcoding the input to the auditory-verbal word frame, where the appropriate arithmetic fact could then be retrieved and then the verbal answer produced (e.g., retrieving the answer as “three plus four equals seven” and then verbally producing the answer “seven”).

While the abstract code model and the triple code model differ with respect to the issue of functional vs. representational modularity, they do share a fundamentally additive architecture. That is, any performance differences related to problem format (e.g., faster RTs for problems written in Arabic digits compared to words) simply reflect processes related to encoding. In the context of the abstract code model, such performance differences would be explained as the cost of converting a specific stimulus type (digits or words) into an amodal, abstract semantic representation that can be further fed into an appropriate calculation mechanism. In the context of the triple code model, these performance differences would reflect a cost of converting from one representational format (e.g., verbal, word-based representation) into another format (e.g., visual, digit-based representation). Critically, both models predict that format manipulations do not interact with calculation processes.

As an alternative to such additive models of mental arithmetic, Campbell and colleagues (e.g., Campbell and Clark, 1988; Campbell, 1994; Campbell and Epp, 2004) have argued for an interactive architecture called the encoding complex model. In this model, performance differences due to manipulation of format are posited to stem from a difference in the degree of encoding-retrieval integration. For example, Arabic digits are frequently encountered in the context of calculation, and hence, strong bi-directional pathways are developed between encoding and retrieval of arithmetic facts in this format. However, number words are less frequently encountered, and hence weaker encoding-retrieval connections are formed for such inputs. While sharing some similarities with the additive models described earlier, this model differs in one critical aspect; changes in format are hypothesized to impact both encoding and calculation processes.

Support for such an interactive model has primarily appeared in the form of an interaction between the variables of problem format and problem size. As one of the classic “effects” in mathematical cognition, the problem size effect refers to the finding that responses for small problems (e.g., problems for which the sum of the operands is no larger than 10) are significantly faster than responses for larger problems (Ashcraft, 1992; Zbrodoff and Logan, 2005). One possible reason for the problem size effect is that large problems tend to be solved by procedural strategies, resulting in slower and more error prone responses (Campbell and Xue, 2001). While the exact mechanism underlying the problem size effect is still up for debate, the more salient finding is that the problem size effect is larger for problems presented in word format compared to digit format (Campbell and Fugelsang, 2001; Campbell and Penner-Wilger, 2006). Campbell and colleagues have argued that this interaction between problem size and format implies that format directly impacts calculation processes, providing support for the interactive model.

Nonetheless, recent research has not settled the debate regarding the independence of encoding and calculation in mental arithmetic. On one hand, some researchers have argued that people form notation-independent representations of numbers. For example, Libertus et al. (2007) recorded ERPs (event-related potentials) during a symbolic and nonsymbolic number comparison task. Adults were presented with single numbers (shown either in Arabic digit format or in nonsymbolic dot format) and asked to decide whether each was less than or greater than 15. They found that the amplitude of the P2 component (210–250 ms after stimulus presentation) increased as the distance between the number and the comparison standard decreased. Moreover, this pattern did not differ between number formats. This led Libertus et al. (2007) to conclude that number comparison proceeds via an abstract processing stage that is independent of number format. This finding mirrored previous work by Pinel et al. (2001), who used fMRI to identify regions in the parietal lobes whose activation was highly correlated with semantic properties of numbers (i.e., numerical distance), but invariant as to whether the number was presented in word or Arabic numeral format.

Similar results have been also found in behavioral experiments. For example, Ganor-Stern and Tzelgov (2008) used a size-congruity paradigm to investigate automaticity of numerical processing. In this paradigm, numbers are presented in differing physical sizes; this results in pairs of number symbols in which the physical comparison is congruent with numerical size (e.g., small 2 and large 8) or incongruent (e.g., large 2 and small 8). The usual finding is that incongruent pairs take longer than congruent pairs. This size-congruity effect (e.g., Henik and Tzelgov, 1982) is often taken as evidence for automatic processing of number magnitude. In their experiment, Ganor-Stern and Tzelgov (2008) presented Arabic speakers with number pairs written in two different notations: Arabic and Indian digits. Ganor-Stern and Tzelgov found that even in mixed pairs (one Arabic digit and one Indian digit), there was still a substantial size-congruity effect. They interpreted this result as support for the notion that both notations are automatically converted to a common representation independent of format (see also Ganor-Stern, 2009).

As mentioned earlier, evidence against this additive view of arithmetic processing has been presented by Campbell and colleagues in the form of a substantial problem size by format interaction on response times (e.g., Campbell and Fugelsang, 2001). Some researchers (e.g., Noël et al., 1997) argue that this signature on RTs does not necessarily imply that format has a direct impact on calculation processes. Noël et al. (1997) argued that such an interaction may be the result of encoding differences between digits and words that feed into the output stage, not the calculation stage. Finally, at least one recent study indicates that the interaction between problem size and format may not be as robust as first thought. For example, Megias and Macizo (2016) failed to find an interaction between problem size and format in a mental arithmetic task¹. Taken together, these issues warrant further investigation of the processes in volved in mental arithmetic, as it appears that we still have more to learn about the potential interplay between encoding and calculation. In the sections below, we outline a new approach to investigating this issue, based on modeling distributions of RTs in a mental arithmetic task.

Most of the studies mentioned above have employed a similar approach to analyzing the effects of experimental manipulations on RTs. Namely, for each participant, the collection of RTs for correct trials in each experimental condition is collapsed to one number, usually the arithmetic mean. This collection of means is then analyzed via an analysis of variance to determine the effect, if any, of each manipulation on RTs. Though popular, this approach is suboptimal for two reasons. First, by collapsing RTs by condition to a single numerical summary (e.g., the mean RT), we lose much information about the distribution of RTs. Second, this procedure is usually carried out only on correct trials. As such, RTs and response accuracy are analyzed separately, even though they are not necessarily independent (e.g., the speed-accuracy tradeoff, Schouten and Bekker, 1967; Wickelgren, 1977). In both cases, ease of analysis comes at the price of lost information about the original patterns of RTs.

One solution to this problem is to employ a mathematical model such as an accumulator model, a model for decision processes that posits a continuous uptake of noisy information that continues until the accumulated evidence exceeds a decision threshold, at which point a response is initiated (Link and Heath, 1975; Luce, 1986; Ratcliff and McKoon, 2008; Ratcliff et al., 2016). One advantage of such an approach is that instead of modeling participants' RTs in each experimental condition by a single mean RT, we can fit a model to the entire distribution of RTs for each participant in each experimental condition. This results in finding a set of parameters that not only describes the distribution mathematically, but also are indicative of the underlying cognitive processes. The advantage is that we can then directly test the effects of our experimental manipulations on the cognitive processes, not just the effects on RTs and/or errors. Hence, RTs and errors become for us a proxy to the underlying cognitive processes, not the sole object of study.

One popular example of a widely used accumulator model is the drift diffusion model of Ratcliff and colleagues (Ratcliff and Murdock, 1976; Ratcliff, 1978; Ratcliff and McKoon, 2008; Ratcliff et al., 2016), which describes a two-choice decision task as the result of such a noisy accumulation process. Specifically, a decision process is modeled as a continuous random walk {X_t} with absorbing boundaries 0 and α. This means that the initial term of the walk X₀ begins somewhere between 0 and α (i.e., 0 < X₀ < α), and the walk terminates whenever X_t = 0 (an incorrect response) or X_t = α (a correct response). Moreover, the random walk terms X_t tend to drift toward one boundary or the other. That is, ddtXt is assumed to be normal with mean γ; we refer to γ as the drift rate. Finally, the decision time DT is modeled as the first time t for which X_t hits either boundary; that is, X_t ≤ 0 (an incorrect response) or X_t ≥ α (a correct response). The total response time RT is then expressed as RT = DT + θ, where θ represents the nondecision component of RT (e.g., stimulus encoding and motor execution).

Modeling RT distributions via this diffusion process results in a set of parameters that can be mapped onto underlying latent cognitive processes. The interpretation of these parameters as indices for cognitive processes has been the subject of much investigation over the past 40 years (see Ratcliff and McKoon, 2008, for a review). Though the full Ratcliff diffusion model results in 7 such parameters (Wagenmakers et al., 2007), for simplicity we restrict our discussion to the following three parameters: α (boundary separation), γ (drift rate), and θ (nondecision time). The boundary separation parameter α represents response caution; a high value of α means that more evidence needs to accrue before a decision can be made. In other words, large values of α reflect conservative decision criteria, whereas small values of α reflect more liberal decision criteria. The drift rate parameter γ represents the quality of information provided by the stimulus. Larger drift rates reflect unambiguous stimuli, resulting in quicker decisions. Smaller drift rates reflect ambiguous stimuli, resulting in longer decision times. Finally, the nondecision time parameter θ reflects encoding and response processes; large values of θ reflect slower encoding and/or execution, whereas small values of θ reflect fast encoding and/or execution.

While fitting RT distributions with these parameters results in a much more detailed description of the underlying RT distributions than using the mean alone, it is not always possible to fit such a model to experimental data. For instance, one problem with the Ratcliff diffusion model is that it is not well suited to tasks with very low error rates (Anders et al., 2016). Consequently, tasks in which participants perform quite well are not fit well by the diffusion model parameters. However, an alternative accumulator model called the shifted Wald model may be well suited to such situations (Carpenter and Williams, 1995; Schwarz, 2001; Heathcote, 2004). The shifted Wald model is a model of RTs based on the Wald (1947) distribution, which represents the density of first passage times of a continuous diffusion process that drifts toward a single absorbing boundary. Mathematically, the probability density function of the shifted Wald model is given by:

where α represents the height of the single response boundary, γ represents the drift rate, and θ represents a positive (rightward) shift of the entire distribution. As proxies for cognitive processes, each of these parameters has a straightforward interpretation similar to (but not quite equivalent; see Matzke and Wagenmakers, 2009) that of the Ratcliff diffusion model (Schwarz, 2001; Heathcote, 2004; Anders et al., 2016): drift rate γ reflects task difficulty via quality of information; response threshold α reflects response caution (amount of information required before response initiation), and θ reflects the nondecision time (e.g., encoding and response processes not directly related to the post-encoding decision process).

Aside from the measurement advantages of modeling RTs via distributions rather than via single point means, the shifted Wald model has further methodological and theoretical advantages in the context of mathematical cognition. First, as error rates in mental arithmetic tasks tend to be quite low, fitting RT distributions with the full Ratcliff diffusion model will be difficult. Instead, we can use the shifted Wald model to describe RT distributions for the correct responses. Further, compared to the Ratcliff diffusion model, the shifted Wald model can be fit with a relatively small number of experimental trials. For example, Anders et al. (2016) found that shifted Wald parameters can be recovered for as few as 50 observations per experimental condition. Finally, as the parameters of the shifted Wald distribution have specific cognitive interpretations, systematic variation in these parameters as a function of a stimulus manipulation (e.g., problem format) can tell us the exact locus of the effect. As described above, it is well known that problem format has an effect on RTs – problems in word format take longer to solve than problems in digit format. However, it is unclear whether this RT effect is localized to the encoding stage, the calculation stage, or both. By modeling participants' RT distributions via shifted Wald models, we can obtain a measure of how format affects each of the parameters α, γ, and θ. In the present study, we are mainly concerned with the question of whether problem format affects calculation. As such, we can make solid predictions about the effects of our manipulations on the drift rate γ, which we assume reflects the calculation process. This assumption comes from the idea that γ is a parameter related directly to a decision process, which in this experiment is a decision about the truth of a proposed addition equation. Critically, if format affects drift rate γ, this will give evidence that format also has a direct effect on calculation over and above the previously established effects of format on encoding. Such a result would favor the interactive encoding complex model over an additive model (e.g., abstract code model or triple code model), where effects of format are isolated to only the encoding stage. Note that at present, clear predictions cannot be drawn regarding the effects of our manipulations on α and θ, so for the purposes of this study, we will focus on the drift rate γ.

There were two main goals in the present study. First, we sought to extend previous work in mathematical cognition by replicating the arithmetic verification task of Campbell and Fugelsang (2001) and applying an accumulator model (the shifted Wald model) to model the resulting RT distributions. Whereas many mental arithmetic experiments use a production task, a verification task is advantageous for this modeling approach. In addition to being the task used in Campbell and Fugelsang (2001), the verification task allows us to measure mental arithmetic processes in the framework of a two-choice decision task, which is the framework employed in most studies that employ accumulator models to study RTs. Second, we aimed to use the results of this modeling to test between two competing models: an additive model, where the stages of problem encoding and answer calculation are functionally independent and effects of problem format are isolated to the encoding stage only, and an interactive encoding-complex model, where effects of problem format are spread between both the encoding stage as well as the answer calculation stage.

Twenty undergraduate students (15 female, mean age = 25.2 years, age range = 19–60 years) participated in this experiment in exchange for partial course credit in their psychology courses. The experiment was reviewed and approved by the institutional review board at Tarleton State University.

Our stimuli were adapted from Campbell and Fugelsang (2001). Each participant completed 288 experimental trials, consisting of four repetitions of a block of 72 single-digit addition verification problems. We manipulated both problem size and problem format. On even numbered trials, the problems were presented in word format using lower case English words (e.g., “five + seven = twelve”). On odd numbered trials, problems were presented in Arabic digit format (e.g., “5 + 7 = 12”). All problems (regardless of format) were composed of operands between 2 to 9, resulting in a set of 36 problems ranging between 2 + 2 = 4 and 9 + 9 = 18. Note that this assumes that commuted pairs such as 2 + 6 and 6 + 2 are counted as one problem. For each commuted pair, both operand orders were presented with equal frequency throughout the experiment. The order of the operands for each problem was alternated across blocks. Within each of the four blocks, each of these 36 problems was presented once in digit format and once in word format. Problem size was defined in terms of the product of operands: small problems had product operands less than or equal to 25, whereas large problems had operands greater than 25. Within each block, half of the problems of each problem size were presented with the smaller operand first; for the remaining half, the larger operand was presented first.

We further manipulated the truth value of each addition problem. Within each set of 36 problems, half were presented as true equations (e.g., “2 + 4 = 6”) and half were presented as false equations (e.g., “2 + 4 = 7”). Across all four blocks, each addition problem was tested in each format twice in a true equation and twice with a different false answer. False answers were generated pseudo-randomly to be within ±4 of the correct answer and never corresponded to either the difference or the product of the operands. Within each set of false answers, each of the numbers 4–18 (i.e., the range of true answers) occurred at least once but no more than four times.

All stimuli were presented using Superlab 5.0 (Cedrus Corporation), appearing as white characters against a black background. Responses were recorded using an RB-740 USB response box with ±2 ms timing accuracy. The experiment was run on a 21.5-inch iMac desktop computer with 1,024 × 768 screen resolution. Text was displayed in 36 point Lucida Grande font. In each problem, the two operands were separated by a single space on either side of the addition sign. The answer to be verified appeared simultaneously with the problem operands to the right and after the equal sign (e.g., 2 + 4 = 6). No other characters appeared on the screen.

We counterbalanced two response rules across our participants. Even-numbered participants indicated true responses by pressing the rightmost button of the response box and false responses by pressing the leftmost button. Odd-numbered participants used a reversed response mapping: they indicated true responses with the leftmost button and false responses with the rightmost button. Each participant was instructed to respond quickly but accurately.

Prior to the first block, we gave each participant a practice block, consisting of 12 trials in alternating word and digit format, using the operand 0 or 1 paired with 6 randomly selected digits ranging from 0 to 9. At the beginning of each trial, a fixation cross appeared at the center of the screen. When ready to begin, each participant initiated the presentation of the equation with a single button press. The fixation dot flashed for 1 s and was then replaced with one of the 72 addition verification problems. Timing began with the presentation of this equation and ended as soon as the participant pressed a button indicating whether the problem was true or false. After each response, feedback was given in the form of a green C (for correct trials) or red E (for incorrect trials), displayed in the center of the screen for 300ms. After feedback, the fixation cross reappeared, signaling to the participant that the next trial was ready to be initiated. After each block of 72 trials, participants were given an opportunity for a short rest. At the conclusion of the fourth block (288 trials completed), the experiment ended and participants were thanked for their participation.

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. The reviewer KL and handling Editor declared their shared affiliation, and the handling Editor states that the process nevertheless met the standards of a fair and objective review.