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INDIVIDUAL DIFFERENCES IN THE DEVELOPMENT OF SKILL IN

MENTAL ADDITION: INTERNAL AND EXTERNAL VALIDATION OF

CHRONOMETRIC MODELS

KEITH F. WIDAMAN AND TODD D. LITTLE

UNIVERSITY OF CALIFORNIA, RIVERSIDE

DAVID C. GEARY

UNIVERSITY OF MISSOURI, COLUMBIA

PIERRE CORMIER

UNIVERSITt DE MONCTON

ABSTRACT: Two studies were undertaken to investigate the development of skill in mental addition. In Study 1, a sample of 123 second-, fourth-, and sixth- graders were administered 140 simple addition problems in a true-false reaction time (RT) format; scores on eight subtests of the Stanford Achievement Test were also available on these subjects. In Study 2, the sample consisted of 63 second-, fourth-, and sixth-graders and 100 college students. The RT tasks used in Study 2 comprised 80 simple addition problems and 80 complex addition problems, and a set of six paper-and-pencil tests for Numerical Facility and Perceptual Speed was also administered to the subjects. The results of the two studies support several conclusions. First, there are substantial individual differences in the age at which the transition from counting to retrieval strategies occurs. A large proportion of students in second grade already rely primarily on retrieval to respond to addition problems, yet a sizeable minority of sixth graders and even college students still apparently rely heavily on counting processes to solve addition problems. Second, there are different rates of automatization for the several processes that underlie skill in mental addition, processes of encoding of digits, recomputing or retrieving the correct sum, and carrying to the next col-

Direct all correspondence lo: Keith F. Widaman, Department of Psychology, University of California, RiversIde, CA 92521.

Learning and Individual Dillerences, Volume 4. Number 3. 1992, pages 167-213. Copyright Q 1992 by JAI Press, Inc. All riohts of reomduction in anv form reserved. ISSN: 1041-6080

168 LEARNING AND INDIVIDUAL DIFFERENCES VOLUME 4, NUMBER 3. 1992

umn. Third, individual differences in efficiency in mental addition, indexed by retrieval speed for retrieval subjects and incrementing speed for digital subjects, exhibited strong relationships with achievement and ability measures. The results of these two studies demonstrated (a) the internal and external validity of the component processes in the proposed model of mental addition performance and (b) the need to use individual-level data to characterize properly the development of the cognitive skills underlying mental addition.

A considerable body of recent research (e.g., Ashcraft 1982; Cornet, Seron, Deloche, & Lories 1988; Siegler 1988a; Widaman, Geary, Cormier, & Little 1989) has investigated the cognitive processes that underlie responses to simple and complex addition problems and the course of development of these cognitive skills. Using chronometric techniques, researchers typically have investigated which of a small set of cognitive processing models best fits average reaction time (RT) data for subjects at each of two or more age levels. The general conclusion supported by chronometric research is that children switch from slower, reconstructive (i.e., counting) strategies to faster, more efficient memory network retrieval strategies during the third or fourth grade (e.g., Ashcraft & Fierman 1982; Cooney, Swanson, & Ladd 1988). For example, Ashcraft and Fierman reported that about one-half of third graders used a counting process to solve addition problems, whereas the remaining third graders and all fourth and sixth graders appeared to use a memory retrieval strategy when responding to the simple addition problems. Cooney et al. reported a similar trend for the development of strategies used in mental multiplication. Specifically, third grade subjects appeared to use algorithmic, counting strategies with multiplication problems, whereas fourth graders tended to retrieve answers from a stored network of multiplication facts.

PROCESS MODELS FOR MENTAL ADDITION

A Verification-Task Model for Simple Addition. In order to contrast alternative cognitive processing models against one another, a basic model representing the processes underlying performance in a domain must be available. A basic model capable of representing the manner in which subjects respond to simple addition problems in a verification task paradigm is presented in Figure 1. The verification task paradigm is the most commonly used procedure and consists of the presentation of an addition problem with a stated sum, which may be either correct or incorrect. Under such presentation, subjects are presumed first to encode the two single-digit addends and the stated sum. Evidence suggests that persons engage in a quick, global evaluation of the approximate correctness of the stated sum (Ashcraft 1982). If the stated sum is grossly inaccurate, the “No” branch is taken, further processing is short-circuited, and a response signifying incorrectness of the stated sum is executed. If the stated sum is approximately

DEVELOPMENTOFSKILLIN MENTAL ADDITION 169

FIGURE 1

Flow diagram of a model for verification task performance on simple addition problems.

Yes’ z I gj 0

Search/Compute Correct Sum: x

Reactivate Stated

Select and Execute Response

correct, processing continues to determine the correct sum precisely. Usually, subjects take the “Yes” branch and then either compute the correct sum of the two digits, search for the correct sum in a memory network of addition facts, or invoke procedural rules or heuristics to determine the correct sum. Occasionally, the “Yes” branch is taken if aspects of the problem lead to interference. For example, interference may occur if the stated answer is incorrect for addition, but is correct for another operation, as in “2 + 3 = 6.” Interference under such conditions typically leads to a longer RT for computing or retrieving the correct sum.

Upon arriving at the correct sum, subjects must decide whether the stated sum is correct or incorrect. Evidence exists (see Widaman et al. 1989) that the digits comprising an incorrect sum must be reactivated or re-encoded, due per- haps to decay of the representation of these digits in working memory, before a decision may be made regarding correctness of the stated sum. Reactivation of the digits in correct sums may also be necessary; however, the extra processing step in Figure 1 reflects the finding that reactivation of digits in incorrect sums is

170 LEARNING AND INDIVIDUAL DIFFERENCES ‘MLIJME 4. NUMBER 3, 1992

more time-consuming than is reactivation of the digits in correct sums (Wid- aman et al. 1989). After deciding on the correctness of the stated sum, subjects select and execute a response consistent with their decision.

A General Componential Model for Mental Addition. Recently, Widaman et al. (1989) presented a more encompassing flow diagram for a componential model representing verification task performance on mental addition. The componential model developed by Widaman et al. is an elaboration of the model in Figure 1, allowing the figural representation of the flow of processing for addition problems with more than two addends and for addends with more than a single digit. Importantly, this componential model also explicitly provides for the process of carrying from one column to the next. The Widaman et al. componential model enables direct translation of the hypothesized processing components for a problem (e.g., encoding of digits, searching for or computing the sum) into structural variables that should relate to RT. Using a sample of college subjects, Widaman et al. found that similar regression models, with fairly stable parameter estimates, provided the best fit to RT data across four types of addition problems that differed considerably in problem complexity. The strong con- vergence between the general componential model and the associated statistical models provided clear support for the Widaman et al. model. For a more complete description of the componential model, see Widaman et al. (1989, pp. 903-906).

COGNITIVE PROCESSES INVOKED DURING THE SEARCH/COMPUTE STAGE OF PROCESSING

Given a process model for simple or complex addition, an important task involves specifying regression models representing alternative cognitive processes or strategies used during particular stages of processing and then contrasting the fit of the regression models against one another. A central goal of this activity is to identify the cognitive operations invoked during the Search/Compute stage of processing. For example, alternative conceptual models may be developed from the basic model for simple addition in Figure 1 by postulating different cognitive processes that subjects might use during the Search/Compute stage. For each hypothesized process, a predictor variable, or structural variable, is specified of which RT should be a linear function. Such regression modeling of RT data is often termed the internal validation of a cognitive processing model. Internal validation of a model is a first and important step in model testing and is established by showing that the model has acceptable parameter estimates and acceptable fit to the RT data that the model was formulated to represent. All other things being equal (e.g., all regression weights falling within bounds enabling interpretation), the regression model with the highest level of fit provides the strongest internal validation, or eviden- tial support, for the hypothesized cognitive process represented by the equation.

DEVELOPMENT OF SKILL IN MENTAL ADDITION 171

Digital, or Counting, Processes. The two classes of cognitive operations typically contrasted are digital (or counting) and memory network retrieval processes, with several variants of each. Groen and Parkman (1972; Parkman & Groen 1971) discussed five alternative counting, or recomputing, models, each of which led to the specification of a particular regression model for RT data. Groen and Parkman presumed the existence of a mental counter that could be set at most once and then incremented in a unit-by-unit fashion. Groen and Parkman showed that, depending upon the strategy used, RT might be a linear function of the first addend, the second addend, the larger addend, the smaller addend (or MIN), or the correct sum (SUM) of the two addends. For example, if subjects use an efficient counting strategy, they might set their mental counter always to the larger of the two addends and then increment the mental counter, in a unit- by-unit fashion, a number of times equal to the value of the smaller addend to obtain the correct sum. If this strategy were used on all problems, then RT would be a linear function of the number of times the mental counter is incremented and, therefore, of the smaller addend, or MIN; also, the regression coefficient for the MIN structural variable would provide an estimate of the time taken for each incrementation of the mental counter. Among counting models, the MIN model consistently provides the best fit to RT data for both children and adults (e.g., Parkman & Groen 1971). Given its acceptable parameter estimates, the superior fit of the MIN model supported the internal validity of the MIN model relative to the other counting models considered by Parkman and Groen.

Interestingly, the counting models discussed by Groen and Parkman (1972) may be thought of as internalized and automatized analogs of overt counting strategies that children use when first learning to add. Siegler and Shrager (1984) described four strategies used by 4- and 5-year-olds when responding to addition problems. Some children used a “counting fingers” strategy, in which they held up fingers on each hand to represent the number of units in the two addends and then explicitly counted all of their raised fingers. In the somewhat more automatized “fingers” strategy, children raised fingers to represent the units in each addend, but did not explicitly count their raised fingers. In a yet more automatized and internalized strategy, the “counting” strategy, Siegler and Shrager reported that children either counted aloud or moved their lips while counting silently. The fourth and final strategy enumerated by Siegler and Shrager was termed a “retrieval” strategy, in which the children gave no explicit indication of any strategy use.

Regarding regression models that would best fit RT data for the four strategies identified by Siegler and Shrager (1984), the “counting fingers” and “fingers” strategies would probably be best fit by a SUM structural predictor, as the two strategies typically represent situations in which children count all units. The third strategy, termed the “counting” strategy, might be fit best by either a SUM or MIN structural predictor, depending upon whether children counted all units or “counted on” units of the smaller addend beginning with the value of the larger addend, respectively. Of course, at this rather early stage in the development of skill in addition, other structural predictors would be reasonable candi-


dates, such as a SECOND predictor, if children use a “counting on” strategy, beginning with the units in the first addend and then incrementing a number of times equal to the second addend. Finally, the “retrieval” strategy identified by Siegler and Shrager is a more difficult strategy to match with potential predictors. Siegler and Shrager identified responding as conforming to the “retrieval” strategy if the child displayed no explicit aids prior to responding. But, this lack of explicit aids may have indicated that counting was highly automatized and internalized, not that some true retrieval strategy was being used. If counting were the correct basis for performance under the “retrieval” strategy, then RT might be fit well by counting structural variables, such as the MIN. On the other hand, if children were retrieving the correct answer from a memory network, then other predictors, discussed in the following section, would be better predictors of RT.

Memory Network Retrieval Processes. The second major category of strategy posited to underlie response to addition problems is that of memory network retrieval. Ashcraft and Battaglia (1978) first introduced the notion of an internalized mental table of addition facts as a first approximation of a long-term memory network for these facts. The table was hypothesized to be square and symmetric and to be bounded by two entry nodes that had nodal values from 0 through 9, representing the two addends of a simple addition problem. The nodal values were assumed to be equally spaced, and answers to addition problems were stored at the intersection of nodal values for a given addition problem. For example, the answer “5” was stored at the intersection of values “2” on Node 1 and “3” on Node 2. Ashcraft and Battaglia noted that the SUM of addends would describe retrieval from this memory network if the search process conformed to a city-block metric. But, Ashcraft and Battaglia found that a nonlinear relation held between RT and problem size and that the square of the correct sum (SUM2) was a better predictor of RT than was the SUM variable. To account for this finding, they proposed that SUM2 was consistent with a tabular network that was “stretched” systematically in the direction of larger nodal values. The superiority of the SUM2 structural variable as a predictor of RT was replicated in several studies by Ashcraft and his associates (Ashcraft & Battaglia 1978; Ashcraft & Stazyk 1981; Hamann & Ashcraft 1985).

However, Widaman et al. (1989) showed that the SUM2 predictor was incon- sistent with the symmetric “stretched” table of facts propounded by Ashcraft and Battaglia (1978). Rather, SUM2 was consistent with search, under a city- block metric, through an asymmetric network, with an entry node reflecting the larger addend and a capture node representing the smaller addend. If the square and symmetric properties of the tabular network were retained, but nodal values were stretched in the direction of larger values, as Ashcraft and Battaglia had proposed, then RT should be a linear function of the sum of the squared addends (ADD2 = [ADDEND12 + ADDEND22]) or the sum of some other power of the addends. Finally, Widaman et al. noted that the product (PRODUCT) of the two addends was consistent with the originally proposed tabular network-


a square, symmetric table with equally spaced nodal values-in that the PROD- UCT represents the areu of the network that must be traversed from the (0,O) entry node to the intersection of the nodal values for the problem (cf. Miller, Perlmutter, & Keating 1984). Thus, four primary structural variables have been identified that are consistent with a tabular network of addition facts: SUM, SUM2, ADD*, and PRODUCT.

At least four nontabular structural variables have also been proposed to represent time-consuming search through a memory network of addition facts. Sieg- ler and Shrager (1984) formulated a distribution of ussociations variable, which represented the proportion of times the correct answer for an addition problem was provided by 4- and 5-year-olds. Addition problems with small addends had more peaked distributions of associations of answers, and the distributions of associations became less peaked as addend magnitudes increased. As the peakedness of the distribution is lessened, the correct answer for the problem is less obvious, and a longer search through the stored memory representation is required to arrive at the correct answer. A second nontabular variable is problem difficulty, proposed by Ashcraft, Fierman, and Bartolotta (1984). Ashcraft et al. used data from a study by Wheeler (1939), who implemented a training study of the addition combinations and recorded the proportion of children learning each addition combination during a two-week training session. Addition problems of higher difficulty should require longer search times to verify than problems with lower difficulty. The final two nontabular predictors, frequency of presentation and order of presentation, were proposed by Hamann and Ashcraft (1986). Hamann and Ashcraft reasoned that simple addition problems with smaller addends are presented earlier and more often in textbooks and in elementary school class presentations relative to problems with larger addends. Thus, the memory networks for addition facts for both children and adults may have greater memory strengths for, and therefore easier access to, problems with smaller addends, because of the greater frequency of presentation and/or earlier presentation of these problems.

The nontabular structural variables have at least two advantages over the tabular ones. First, the nontabular variables are consistent with retrieval models that are more similar to models for nonarithmetic types of memory networks, for example, models for semantic memory. If the nontabular models are more ap- propriate representations for memory networks for arithmetic facts than are tabular ones, then research on memory networks for numerical facts may move forward at a faster rate by attempting to generalize basic findings on semantic memory networks to networks for numerical facts. Second, it is possible that nontabular models reflect the true underlying parameters of memory network access and that tabular structural variables are highly related to RT only because of the high correlations between values on tabular and nontabular structural variables (Campbell & Graham 1985; Siegler 1988b). But, as Widaman et al. (1989) noted, the choice between tabular and nontabular models must ultimately be based on empirical considerations, such as which type of model leads to maximal goodness of fit to data and which type of model results in the greatest

174 LEARNING AND INDIVIDUAL DIFFERENCES VOLUME 4, NUMBER 3, 1992

number of unique, testable hypotheses. In the present article, we will offer evidence on the goodness of fit of alternative models to data.

PROBLEMS IN MODELING RT DATA FOR ADDITION PROBLEMS

Lack of Unique Predictive Utility. The determination of solution strategies used for solving numerical problems and of changes in solution strategies as a function of age or grade are important questions to pursue in research on arithmetical skills. However, at least two general types of problems arise when fitting regression models to RT data on addition problems (cf. Pellegrino & Goldman 1989). The first type of problem concerns the lack of unique predictive utility of the various predictor variables used to represent distinct cognitive processes underlying mental addition, a problem having both statistical and more theoretical aspects. With regard to the statistical aspects, several investigators have noted (e.g., Groen & Parkman 1972; Ashcraft & Battaglia 1978; Miller et al. 1984) that many of the predictor variables for alternative conceptual models are highly correlated. For example, correlations among the MIN, SUM, SUM2, and PRODUCT structural variables tend to be greater than .85 (cf. Ashcraft & Battaglia 1978). The high correlations among these predictors ensure that a regression model incorporating one of the predictors (e.g., PRODUCT) will produce statistical results, such as the proportion of RT variance explained by the model, that are rather similar to those produced by regression models incorporating alternative, highly correlated predictors (e.g., SUM2). In a later section, we discuss the way in which’we tested statistically the unique predictive utility of alternative, highly correlated models. But, the high correlations among predictor variables have at least one unfortunate outcome that is difficult to deal with: RT data generated according to one processing model may be better fit by an alternative regression model due to inadvertent sampling fluctuations in RT data (cf. Pellegrino & Goldman 1989). Replication is a workable, if inefficient solution to such a problem; hopefully, future methodological research will provide more efficient solutions.

A related, but more theoretical concern is the uniqueness of the connection between a structural predictor variable and the conceptual model, or cognitive process, that it is presumed to represent. Despite the high correlations among predictor variables, we contend that each structural predictor variable represents a distinct cognitive process relative to others in its class (cf. Groen & Parkman 1972; Miller et al. 1984). For example, the MIN predictor variable represents one particular counting model that is distinct from other counting models with regard to both (a) its conception, or the manner in which the mental counter is used, and (b) the statistical predictions made. On the other hand, it is well known that each of the structural variables associated with one or another of the five counting models is also associated with other types of processing models (cf. Widaman et al. 1989). As a result, empirical evidence for the MIN model may support more strongly the MIN model relative to other count-

DEVELOPMENT OFSKILL IN MENTAL ADDITION 175

ing models, but provides equally strong support for models within other, non- counting classes of models. Research on mental arithmetic is still in a formative state, and the distinctiveness of a conceptual model from others within its class of models may be sufficient to guide useful research investigation. As empirical research in the arithmetic domain progresses, a core goal should be to develop distinctive predictions for each conceptual model both within as well as across classes of models.

Level of Observation. The second general type of problem concerns the correct level of observation in regression modeling of RT data. Here, it is not clear that the typical group-level approach to these questions is an entirely suitable one. Under the typical approach, reaction times for each problem are averaged across all subjects in a particular sample, for example, across all subjects in a particular age or grade group. Then, characteristics of the problems are used to embody alternative cognitive processing models, and the fit of regression models representing the alternative processing models are contrasted to determine which processing model provides the best fit to the data from the sample of subjects. Conclusions regarding the development of arithmetic skill are then drawn at the group level, to support claims that children move from the use of digital or reconstructive processes to memory network retrieval when responding to addition problems (e.g., Ashcraft & Fierman 1982).

Recently, Siegler (1987; 1988a; 1988b), Cooney et al. (1988), and others have begun to question the utility of the typical group-level approach for representing how children solve problems and how children’s solution strategies change with age or grade. For example, Cooney et al. (1988) found that a small proportion of third-graders used memory retrieval when responding to simple multiplication problems, but that most fourth-graders used retrieval. Moreover, Cooney et al. showed that important individual differences were masked or obscured by group-level analyses, which represent an averaging across different strategies at the interindividual level. Going further, Siegler (1987) argued that a child may use more than one strategy across the set of simple addition problems. If a single regression analysis is performed for each child across problems solved by that child using different strategies, this represents an averaging across different strategies at the intraindividual level, and the resulting parameter estimates may be uninterpretable. In the present study, we will present additional evidence that group-level modeling of RT data may lead to inappropriate representations of data.

AIMS OF THE PRESENT STUDY

Previous research has provided evidence that there are important and systematic developments in the skills underlying arithmetic performance during the elementary school years. However, important questions remain to be answered. A first and rather crucial issue is the manner in which the transition from digital or reconstructive processes to memory network retrieval when solving addition

176 LEARNING AND INDIVIDUAL DIFFERENCES VOLUME 4. NUMBER 3, 1992

problems is portrayed. In studies by Ashcraft and associates (1982; Ashcraft & Fierman 1982; Hamann & Ashcraft 1985), the conclusions reached have been fairly general and overarching: at some point in third or fourth grade, children move from digital to retrieval strategies. This movement between strategies has usually been discussed as change that happens for all children in a particular grade. However, it is more likely that the transition between strategies is an individual difference variable, that individuals gradually move from primary reliance on a less mature and efficient strategy to another more mature and efficient one as they become able to execute the more mature strategy successful- ly. To represent this, analyses of data from individual subjects must be conducted, and the results must then be summarized in order to characterize the interindividual development of skill in a domain.

A second important task is the investigation of developmental changes in the efficiency with which the several cognitive processes underlying arithmetic abilities (e.g., encoding of digits, search/compute processes) are executed. Kail (1986, 1988) presented developmental trends for processes underlying a range of cognitive tasks, including mental addition. Of the models considered by Kail, the best-fitting one represented development in cognitive skills as an exponential function of chronological age; Kail claimed that these results supported certain theoretical interpretations of developmental change (e.g., replacement models of learning) relative to others (e.g., accumulation models). But, Horn (1988; Horn & Cattell 1982) and Cattell (1971) have argued that crystallized abilities, of which arithmetic or numerical facility is an exemplar, reflect the degree to which a person has incorporated the “intelligence of his/her culture,” which occurs through systematic influences of acculturation. During childhood and adolescence, these systematic influences of acculturation may be embodied, in large measure, in the standardized educational practices found in schools. Hence, at present, charting the developmental course during childhood and adolescence of improvement in the cognitive skills underlying mental arithmetic leads to at least two interesting research questions: whether the different cognitive skills underlying arithmetic processing have similar or dissimilar developmental trends, and whether developmental improvements in skill during childhood and adolescence are a function of chronological age or grade in school.

Yet a third important question to consider is the external validity of parameter estimates from cognitive processing models; external validity is shown by inves- tigating the relations between RT parameter estimates and variables from out- side the RT task. External validity for the mental addition parameter estimates would be supported if the parameter estimates correlated highly with more traditional ways of assessing arithmetic abilities, using either ability or achievement tests; such relations for mental addition component processes have been little studied. In a sample of college students, Geary and Widaman (1987) found rather strong levels of convergent validity between the parameter estimates for processes unique to mental arithmetic and ability tests of numerical facility. However, determining whether such relations hold at other ages is important as


well. The present study was designed to provide answers to the preceding questions.

STUDY 1

In previous studies of the development of skill in mental addition, the primary focus of analyses has been internal validation of cognitive processing models at the group level. In Study 1, simple addition problems were presented in a verification RT format, and the modeling of the data at both the group and subgroup levels was undertaken. The primary questions investigated were: (a) what proportions of students at each of several grades in elementary school use different strategies with simple addition problems, (b) do group-level analyses misrepresent the development of arithmetic skills, (c) what are the age-related trends in efficiency of executing each process, and (d) do the parameter estimates based on cognitive processing models predict achievement test scores?

METHODS

SUBJECTS

The total sample for Study 1 consisted of 123 elementary school students (63 males) from grades 2, 4, and 6. There were 37 (21 males) second-grade subjects, 45 (22 males) fourth-grade subjects, and 41 (20 males) sixth-grade subjects. The mean ages of the second-, fourth-, and sixth-grade subjects were 7.6 years (SD = .64), 9.7 years (SD = .60), and 11.7 years (SD = .69), respectively.

CHRONOMETRIC PROBLEM SET

Addition Problems. Evaluation of the skill in mental addition was based on RTs for each subject as they responded to simple addition problems. The addition problems presented consisted of 70 of the basic 100 simple addition problems used by many previous researchers (e.g., Ashcraft & Battaglia 1978; Groen & Parkman 1972; Widaman et al. 1989). The problems were selected representatively such that the digits O-9 occurred equally often in the augend position and equally often in the addend position. To create the addition combinations of augend with addend, the digits were classified as small (O-3), medium (4-6), and large (7-9) and randomly balanced such that a small augend occurred once each with a small, medium, and large addend. Similarly, these constraints applied to “me-

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dium” and “large” addends. As only 70 of the 100 simple addition problems were used, only 7 tie problems were allowed to occur. Each of the 70 addition problems was presented twice, once matched with a stated sum that was correct and once with an incorrect stated sum. Incorrect problems were randomly generated such that one-half differed from the correct sum by -+-1, the other half by ?2. Across correct and incorrect problems, 140 simple addition problems were presented.

Addition problems were presented in a random order having three constraints: (a) no two identical stated sums occurred in sequence (whether correct or incorrect); (b) no more than three correct or incorrect problems occurred in sequence; and (c) no two identical digits, except ties (e.g., 3 + 3), appeared in two sequential problems as augend, addend, or stated sums. These constraints were adopted to reduce any priming or other biasing effects.

Apparatus and Procedures. All addition problems consisted of two single-digit integers and a stated sum. The problems were presented vertically and centered on a 30 cm by 30 cm computer screen in four sets of 35 problems, with a pause of approximately 2 minutes between each set. A practice set of 5 correct and 5 incorrect problems, selected from the 30 correct and 30 incorrect problems not included in the test stimuli, was presented prior to the first set. For all problems, each presentation sequence was initiated by a centered “READY” prompt of 500 ms duration followed by a 1000 ms clear screen pause; a problem then appeared and remained until a response was made. Succeeding a correct response, a 1000 ms blank screen pause occurred, which was followed by the “READY” prompt for the next addition problem. Succeeding an incorrect response, a “WRONG” prompt of 1000 ms duration occurred, followed by the 1000 ms blank screen pause, which was in turn followed by the next “READY” prompt.

Tested individually in a quiet room, subjects were seated approximately 70 cm in front of the computer screen. Subjects were instructed to determine as quickly and accurately as possible whether the stated sum of the problems was correct or incorrect and then to respond on a response board by depressing one button to signify that the stated sum was correct or by depressing a second button to signify that the stated sum was incorrect. Subjects were also informed that any problems incorrectly answered would be presented again until she/he answered correctly on each problem. To enable this, a procedure that recursively presented all incorrectly answered problems at the end of each experimental set was included in the computer program that controlled the presentation of the addition problems. By using a Cognitive Testing Station timing card, each subject’s RTs to the 140 simple addition problems were recorded with *l ms accuracy.

ACHIEVEMENT TEST BATTERY

In order to conduct an external validation of the parameter estimates derived from the cognitive processing models, scores for each subject on eight subscales of the Stanford Achievement Test (SAT; Madden, Gardner, Rudman, Karlsen, &


Merwin 1974) battery were obtained from school records. These eight subscales were:

Mathematics Computation (MCOMP). Students decided if a simple mathematical equation (e.g., 4 plus 2) is either greater than, less than, or equal to a second simple equation (e.g., 2 plus 4).

Mathematics Applications (MAPPL). Students were presented with word problems that required simple arithmetic computations and were asked to solve each problem. After arriving at a solution, the student then matched the derived answer with four possible alternative answers or chose an “NH” or “Not Here” alternative.

Mathematics Concepts (MCONC). Students read a statement or question about a math concept (e.g., “which numeral has the greatest value?“) and then selected the best answer from four response alternatives.

Listening Comprehension (LCOMP). Students listened to a passage that was read aloud by their teacher and then listened to a number of questions regarding the passage. For each question, students selected the best answer from four possible response alternatives read aloud by the teacher, marking the best answer on their answer sheet.

Vocabulary (LVOC). The test consisted of a set of sentence fragments, each comprised of the first four or five words of a sentence. The student decided which of four response alternatives best completed each sentence. Both the sentence fragments and the response alternatives were real aloud by the teacher.

Reading Comprehension (RCOMP). Students read a paragraph and were provided with a number of sentences that referred to ideas or concepts presented in the paragraph. The student decided which of four response alternatives best completed a paraphrased idea or concept from the paragraph.

Reading Word Study Skills (RWSS). Students read silently a word that had a phoneme underlined and then selected from four response alternatives the word having the same sound as the phoneme underlined in the target word.

Spelling (SPELL). Students read four phrases in each of which a word was underlined and then identified which underlined word was misspelled given its usage in the phrase (e.g., honey be).

Procedure. The SAT test was administered by teachers during annual achievement testing in the elementary schools. Testing took place approximately three months after the chronometric data were collected. SAT scores were then obtained from school records.

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ANALYSES

Internal Validation. The internal validation of chronometric models was assessed using multiple regression models representing RT to simple addition problems as a function of characteristics of the problems. Because Siegler (1987) has warned against averaging across subjects using different strategies, analyses were performed at both the group (i.e., grade) and individual levels. After categorizing subjects into digital and retrieval process groups, analyses were also performed at subgroup levels. The subgroup-level analyses were performed by averaging across all subjects for a single grade who used a particular strategy (e.g., second-grade digital subjects).

In regression modeling of RT data, the significance of the difference in fit of competing regression models is an important issue. When comparing competing regression models in the present study, the competing models typically had one or more variables in common, differing only in the structural variable specified for the Search/Compute stage. That is, two competing models might have identical predictors except that one equation employed the SUM2 as the predictor for the Search/Compute stage whereas the competing model employed the PROD- UCT. Because neither of the competing models is nested within the other, it is difficult to test statistically the difference in fit of the two models. However, both models would be nested within a model that employed both SUM2 and PROD- UCT as predictors. We could then test whether either of the competing models. explained significantly less variance than did the model containing both predictors. For example, assume that a model with SUM2 had an X2 = .800 and that a model with PRODUCT had an X2 that was .Ol larger, or R2 = .810. Assume also that the BOTH model (i.e., the model containing both SUM2 and PRODUCT as predictors) had an R 2 = .811, which was only slightly larger than that of the better fitting of the two competing models, the PRODUCT model. In such situations, deleting the SUM2 variable from the BOTH model would result in the PRODUCT model, which would differ nonsignificantly from the BOTH model both statistically and practically (i.e., in terms of the change in X2 values). In contrast, deleting the PRODUCT variable from the BOTH model would result in the SUM2 model, leading to a drop of .Oll in the R*, a difference that would be statistically significant and of practical importance as well. In the present study, differences between competing models in the X2 measure of fit of .Ol or greater invariably were statistically significant, and X2 differences less than .Ol were frequently statistically significant. To conserve space, these statistical significance tests were not reported, as most differences between competing models in R2 were .Ol or larger.

In some regression models reported below, we invoked certain forms of constraints across grade levels on the estimated regression weights. These constrained regression analyses were performed for three major reasons, all of which are interrelated. One reason was to provide more well-conditioned estimates of parameters for each grade level. Regression weights are known to fluctuate to a certain extent across samples from a single population. In the


present study, students at different grade levels constituted samples from different populations, populations presumed to differ systematically in expertise on arithmetic. As a result, the unconstrained analyses resulted in parameter estimates that were (a) estimates, more or less accurate, of population parameters for that grade, and (b) estimates that should be systematically related to those at other grade levels. For example, the regression weight for the memory retrieval process at a given grade level should be less than or equal to the regression weight for a previous grade level, reflecting improved, faster retrieval of arithmetic facts from long-term memory storage. The constraints we tested represented the reasonable hypothesis that improvements in arithmetic expertise should be reflected in RT parameter estimates that exhibit smooth, that is, not saltatory, change as a function of increasing expertise with arithmetic, or roughly as a function of grade.

A second reason for pursuing these analyses was to model the developmental trends in performance as conforming to a constrained pattern and then to determine whether the constraints invoked were reasonable. If constraints are reasonable, invoking the constraints should affect the fit of the regression model very little; if constraints are unreasonable, then model fit should be worsened significantly when constraints are invoked. Furthermore, because the constrained regression models were nested within unconstrained models, the difference in fit between the two types of models could be tested for statistical significance. The third reason for employing the constrained analyses was par- simony. Assuming that constraints on parameters were reasonable, the resulting regression models were reduced, restricted parametric representations that fit the data essentially as well as more highly parameterized models having a larger number of parameter estimates. The restricted, or constrained, models therefore were more parsimonious models that should provide a context for future research attempting to replicate and extend the present results.

External Validation. The external validation of parameter estimates based on chronometric models involved relating the parameter estimates to SAT scale scores. If a sufficient number of subjects use a common strategy, the optimal way of modeling these relations is through use of structural equation modeling. Struc- tural models were fit to the data for network retrieval subjects using the LISREL 7 program (Joreskog & Sorbom 1988). Model parameters were estimated using maximum likelihood. Models were evaluated using both statistical and practical indices of fit. The statistical measure of fit was the likelihood ratio chi-square that accompanies maximum likelihood estimation. The practical measures of fit were the rho, p (Tucker & Lewis 1973) and delta, A (Bentler & Bonett 1980) coefficients. According to Bentler and Bonett (1980), p and A values above .90 are adequate, but Tucker and Lewis (1973) suggested that values, for p at least, be above .95 before a model is accepted.

For groups of subjects too small in number to perform structural modeling, multiple regression analyses were performed to represent the relations between chronometric model parameters and SAT scale scores.


RESULTS

Following typical procedures with chronometric data (cf. Ashcraft 1982), only the correct responses were included in the analyses reported in this article. However, with the recursive procedure employed, RTs associated with correct responses for each of the 140 problems were obtained for each subject. Outliers were identified and removed if an RT was beyond the .Ol level of the conditional distribution of responses for a saturated regression model. The saturated regression model included as predictors structural variables representing encoding of digits, re-encoding of digits in incorrect sums, the MIN and PRODUCT structural variables, and an intercept difference between true and false problems (these predictors are described below). Only 2.3% of the responses were identified as outliers, an outlier rate similar to those reported in other recent RT studies of mental addition (e.g., Ashcraft & Battaglia 1978; Widaman et al. 1989). Consistent with previous research in this area (e.g., Ashcraft & Battaglia 1978; Widaman et al. 1989), only responses to nontie problems were used to model the processing strategies used by subjects.

INTERNAL VALIDATION OF CHRONOMETRIC MODELS

Group-Level Analyses. Using multiple regression techniques (Cohen & Cohen 1983), alternative information processing models were specified and then fit to the average RT for each of the 126 nontie problems across all subjects at a given grade level. The search/compute processing models estimated included the five counting based models proposed by Groen and Parkman (1972) and the seven additional network retrieval models discussed earlier (described in more detail by Widaman et al. 1989). Based on previous research, we hypothesized (a) that the MIN structural variable would be the best predictor among the counting models (e.g., Groen & Parkman 1972), and (b) that the product of the two addends (PRODUCT) would be the strongest predictor among retrieval structural variables (e.g., Widaman et al. 1989). Furthermore, we predicted (c) that the group-level models would replicate previous findings, showing that children move from digital strategies in early grades to retrieval strategies in later grades.

In addition to the search/compute processing model, a truth parameter (TRUTH, coded 0 for correct problems and 1 for incorrect problems) was fit with each equation. The TRUTH parameter allows for an intercept difference in RT between true and false problems, which reflects a difference in response selec- tion and execution for the two types of problem (see Widaman et al. 1989).

Widaman et al. (1989) also found that time to encode the digits in addition problems could be estimated with adequate precision. Therefore, one parameter (ENCODE) was specified to represent the number of digits encoded in the problem. The ENCODE parameter had values of 3 for problems in which a single-digit answer was supplied and 4 for problems in which a double-digit answer was provided. A second parameter, REENC, represented a re-encoding


of the digits in incorrect stated sums. Hence, REENC took on values of 0 for correct sums, 1 for single-digit incorrect sums, and 2 for double-digit incorrect sums. Based on the findings of Widaman et al. (1989), the rate of re-encoding digits in incorrect sums was constrained to be identical to that for the original encoding of the digits in the problem. Judging the goodness of fit for each equation was based on the overall model X2 and the significance of the unique or independent variance explained by each structural variable (Cohen & Cohen 1983).

Group-level analyses of average RTs to the 126 non-tie simple addition problems are reported in Table 1 and supported previous conclusions regarding development of skill in arithmetic. Specifically, digital or counting models best represented reaction time for second graders, reconstructive and memory retrieval models fit approximately equally well for fourth graders, and memory retrieval models were clearly best for sixth graders. Among counting structural variables, the MIN variable was the best predictor of RT at each of the three grade levels; the best alternative counting variable was the SUM, which explained approximately 5 percent less variance than did the MIN variable. Among network retrieval structural variables, the PRODUCT variable explained

TABLE 1

Summaries of Group Regression Analysis for Study 1, By Grade

Equation R2 F df RMSE

Grade 2 (N = 37, Ti?: = 5 184 ms) RT = 1929 + 301 (ENCODE) + 771.5 (MIN) ,863 386.32 2,123 751.6

+ 301 (REENC)

Partial F: 17.20, 463.57, 17.20 RT = 2403 + 252 (ENCODE) + 91.9 (PRODUCT) .845 335.19 2,123 798.6

+ 252 (REENC)

Partial F: 10.28, 396.61, 10.28

Grade 4 (fv’ = 45, ??f = 2940 ms) RT = 1239 + 175 (ENCODE) + 374.5 (MIN) ,883 465.27 2,123 339.8

+ 175 (REENC)

Partial F: 28.46, 534.47, 28.46 RT = 1479 + 147 (ENCODE) + 45.1 (PRODUCT) ,877 439.55 2,123 348.5

+ 147 (REENC)

Partial F: 18.26, 502.37, 18.26

Grade 6 (IV = 41, E = 2305 ms) RT = 959 + 172 (ENCODE) + 241.7 (MIN) .796 240.66 2,123 327.0

+ 172 (REENC) Partial F: 29.70, 240.38, 29.70 RT = 1135 + 144 (ENCODE) + 30.1 (PRODUCT) .820 279.32 2,123 307.9

+ 144 (REENC) Partial F: 22.66, 286.87, 22.66

Note: All models significant at the p < .OOOl level. All partial F-ratios are significant at the p i .Ol level.

184 LEARNING AND INDIVIDUAL DIFFERENCES MLUME 4. NUMBER 3, 1992

more variance at all three grade levels than did any other predictor. The best alternative network retrieval structural variable, the difficulty measure from the Wheeler (1939) study, explained approximately 4 percent less variance than did the PRODUCT variable. As shown in Table 1, structural variables representing the Search/Compute stage of processing (i.e., MIN and PRODUCT) explained significant proportions of RT variance at each grade level, as did predictors representing encoding of digits and the re-encoding of the digits in incorrect stated sums. The intercept difference between true and false problems, represented by the TRUTH structural variable, was nonsignificant at each grade level.

Subgroup-Level Analyses. All regression models discussed in the preceding section were also fit to the data for individual subjects. Interestingly, at the individual subject level, over 50 percent, or 21 of 37, of the second graders appeared already to be using memory network retrieval when responding to the addition problems, based on the superior fit of regression models representing retrieval process models. Conversely, almost 20 percent, or 8 of 41, sixth graders still apparently utilized reconstructive, counting processes. Thus, the group-level results presented in Table 1, by averaging across all subjects at each grade level and thereby across subjects using different strategies, may represent a biased picture of the acquisition of skill in mental addition. In other words, the group- level analyses may fail to represent accurately the way in which individuals acquire expertise in numerical facility.

To circumvent the preceding problem, subgroups of individuals at each grade level were identified. At each grade level, one group of subjects was identified as digital subjects, based on the superior fit of the MIN structural variable to their RT data; a second group of subjects was identified as retrieval subjects, based on the superior fit of the PRODUCT variable to their RT data. Then, the average RT to each problem for each subgroup of subjects was calculated and used as the dependent variable in a series of analyses. As shown in Table 2,40 subjects were identified as digital subjects, with 16, 16, and 8 students so identified in grades 2, 4, and 6, respectively. Conversely, 83 subjects were identified as using retrieval to respond to simple addition problems, with 21, 29, and 33 students in the retrieval subgroups at the second, fourth, and sixth grade levels, respectively.

Two forms of regression analysis were then performed. First, unrestricted analyses were performed for each of the six groups, analyses in which regression estimates were freely estimated for each structural variable for each group. Then, constrained analyses were estimated in the following manner. Consider- ing the digital subgroups of subjects in the second, fourth, and sixth grades, if there was a fairly consistent change with grade in a particular parameter estimate (e.g., the regression estimate associated with the MIN variable), then the developmental trend was constrained to follow either (a) a linear decrease as a function of grade or (b) a log-linear decrease in which the change from one grade level to the next was a negatively accelerated function, such that the change from Grade 4 to Grade 6 was one-half as large as the change from Grade 2 to

DEVELOPMENT OF SKILL IN MENTAL ADO/T/ON 185

TABLE 2

Summaries of Constrained Subgroup Regression Analysis for Study 1, By Grade

Equation

Unrestricted

Estimation

R2 RMSE

Consrrained

Esrimafion

Ce = 0 T& # 0

R2 R2

Digital Subjects (N = 40, m = 3926 ms)

Grade 2 (N = 16, E = 6042 ms)

RT = 2278 + 166 (ENCODE) + 1167.4 (MIN) ,838 1178.3 .837 .837

+ 166 (REENC)

Grade 4 (N = 16, E = 3367 ms)

RT = 1461 + 166 (ENCODE) + 524.6 (MIN) .860 484.9 ,857 ,844

+ 166 (REENC)

Grade 6 (N = 8, m = 2370 ms) RT = 1052 + 166 (ENCODE) + 203.2 (MIN) ,710 390.7 ,703 ,685

+ 166 (REENC)

Network Retrieval Subjects (N = 16, E = 3192 ma)

Grade 2 (N = 21, ??f = 4599 ms)

RT = 1815 + 394 (ENCODE) + 60.5 (PRODUCT) ,782 741.1 ,782 ,782

+ 394 (REENC)

Grade 4 (N = 29, E = 2681 ms)

RT = 1419 + 121 (ENCODE) + 40.3 (PRODUCT) .861 320.4 .859 ,859

+ 121 (REENC)

Grade 6 (N = 33, E = 2296 ms)

RT = 1221 + 121 (ENCODE) + 30.2 (PRODUCT) ,807 323.6 ,806 ,806

+ 121 (REENC)

Nore: All models significant at the p < .COOl level. All partial F-ratios are significant at the p < .Ol level.

Grade 4. The better fitting of the linear or log-linear trends was then used to constrain estimates across equations. If no clear developmental trend was in evidence, then equality constraints across grade levels were employed. Identical procedures were followed in representing the data for the retrieval subgroups.

The results of constrained subgroup analyses are presented in Table 2. The results for the digital subjects are presented in the top half of Table 2. One equation is shown for each of the three grade levels. For each grade level, the fit of the regression model (i.e., the squared multiple correlation, R2, and the root mean square error, Rh4SE) under unrestricted estimation is provided along with the fit of the models under constrained estimation (i.e., R* estimates, explained in more detail by Widaman et al. 1989). Comparing the variance explained under unrestricted and constrained estimation, it is clear that little in the way of explained variance is lost under constrained estimation. Given the comparable figures under unrestricted and constrained estimation, only the estimates from constrained models are presented in Table 2.

Turning to the parameter estimates for the digital subjects, the estimates for

186 LEARNING AND INDIVIDUAL DIFFERENCES VOLUME 4. NUMBER 3, 1992

both the intercept and the MIN variable conformed most closely to the log-linear trend as a function of age. That is, the intercept dropped over 800 ms from second to fourth grade, and then dropped half that amount, or about 409 ms, from fourth to sixth grade. Likewise, the time for incrementing the mental counter, indexed by the MIN regression weight, dropped over 640 ms from second to fourth grade and then dropped by half that amount, or 321 ms, from fourth to sixth grade. In contrast, an equality constraint provided optimal fit for the encoding and re-encoding processes, suggesting a lack of age-related changes for these processes.

The constrained regression analyses of data from the retrieval subjects are presented in the bottom half of Table 2. As with the data from the digital subjects, these data revealed that little loss in explained variance accompanied the constrained estimation of regression weights. As a result, only the constrained estimates of regression parameters are presented in Table 2. The intercept estimates again showed a log-linear trend as a function of age, decreasing by almost 400 ms from second to fourth grade and then by half that amount from fourth to sixth grade. The efficiency of retrieval from a stored network of facts, represented by the PRODUCT parameter estimates, also revealed a nonlinear trend with age, dropping by 20 ms between the second and fourth grades and then by another 10 ms from the fourth to sixth grade levels. For the retrieval subjects, neither the linear nor the nonlinear trends provided a reasonable representation of the change in estimates of encoding time. Instead, a rather large, freely estimated parameter was obtained for subjects in Grade 2, and an equality constraint across estimates at Grades 4 and 6 was acceptable.

EXTERNAL VALIDATION OF MODEL PARAMETERS

Subjects Using Retrieval Processes. For the 83 subjects who were identified as retrieving the correct answer for addition problems, covariances among their chronological age, their processing model parameter estimates, and their achievement test scores were calculated. Structural equation models were fit to the covariance matrix to represent the relations among reaction time parameter estimates and achievement test scores. As shown in the top half of Table 3, the null model, representing the hypothesis of null relations among all variables, can be rejected (p < .OOOl).

The second model considered, termed Model 1, was one in which the PROD- UCT parameter estimate and the standard deviation of each person’s residual reaction times (or RMSE) were used as indicators of an Addition Efficiency latent variable, intercept and encoding speed estimates were combined and served as the indicator of a Speed latent variable, the three mathematics tests from the SAT were indicators for a Mathematics Achievement latent variable, the Listening Comprehension and Vocabulary tests served as indicators of a Verbal Com- prehension or Verbal Achievement latent variable, and the remaining three tests were indicators of a Reading Skills latent variable. Direct paths from the Addi-

187

TABLE 3 Indices of Fit for Structural Models Relating Information Processing Parameter Estimates and SAT

Scores in Study 1: Subjects Using Retrievai Processes

Chi-Square Test

Mode! Value df Prob rho d&a

Fit of alternative models

0. Null model 1414.52 78 <.OfIOl - -

1. Model 1: Simple model postulating two 101.49 5.5 <.OOOI .951 ,928

information processing and three

achievement latent variables

2. Model 2: Model 1, plus three other factor 63.50 52 .132 ,987 .955

loadings on achievement factors

Differences in fit of alternative models

Model 0 vs Model 1 1313.03 23 C.0001 .951 ,928

Model 1 vs Model 2 38.78 3 <.OOiJl ,036 .027

tion Efficiency factor to each of the three achievement latent variables were specified a priori; paths from the Speed latent variable to the achievement latent variables were to be added as required by the data. In addition, the linear (AGE) function of chronological age was partialled from all latent variables, and the quadratic (AGE2) function of age was additionally partialled from the Addition Efficiency latent variable. Model 1 provided a highly significant improvement in fit over the null model (see bottom half of Table 3), but was rejected on both statistical (p < .OOOl) and practical grounds, as the A measure of fit was less than .95.

Three additional loadings were added to Model 1 in order to arrive at Model 2. Specifically, direct paths from the Verbal Comprehension factor to the Mathe- matics Concepts and Mathematics Applications tests were specified to account for the verbal variance in the two tests, as both of the latter tests consisted of verbally stated math problems. Given this respecification, the Mathematics Achievement latent variable represented a somewhat more narrowly defined numerical computations construct. The other direct path added was from the Verbal Comprehension factor to the Reading Comprehension test, to reflect the comprehension aspects of the latter test. Model 2 was acceptable statistically, x2 (52) = 63.50, ~7 = .132, as well as practically, with p = .987 and A = .955.

The standardized parameter estimates from Model 2 are shown in Figure 2. All common factor loadings were highly significant, falling at least 4 standard errors from zero. Moreover, the unique variances for the measured variables tended to be rather small, indicating that the latent variables explained the majority of the variance of the measured variables.

Turning to relations among the latent variables, the direct path from the Addi-

188 LEARNING AND lNDlVlDlJAL DIFFERENCES VOLUME 4. NUMBER 3, 1992

FIGURE 2

Structural relations among the information processing and achievement variables and chronological age (Observed variable codes: 1 = Chronological age (in years), 2 = Age squared,

3 = Product parameter estimate, 4 = Root mean squared error, 5 = Intercept + encoding speed, 6 = Mathematics Computation, 7 = Mathematics Concepts, 8 = Mathematics Applications, 9 = Listening Comprehension, 10 = Vocabulary, 11 = Reading Comprehension, 12 = Reading and

Word Study Skills, 13 = Spelling).

tion Efficiency factor to the Math Achievement factor was quite sizable (p = -.59; partial Y* = .49), indicating that the degree of automatization of the cognitive processes underlying mental addition was strongly related to mathematics achievement during elementary school. The relationship between the Addition Efficiency factor and Verbal Comprehension was smaller (/3 = -.42; partial y2 =

.22), reflecting a weaker relation between these two factors. Interestingly, there was a moderate relationship between the Addition Efficiency factor and the Reading Skills factor (p = -.56; partial Y * = .34), suggesting that persons who have more highly automatized memory networks for addition facts also have faster access to memory representations for the written form of words.

Subjects Using Digital Processes. For 40 of the 123 students, a counting model provided a better fit to their data than did a memory retrieval model. The structural variable that provided the best fit for each of the 40 students was MIN. This result is consistent with the hypothesis that the students set a mental counter to the value of the larger addend and then incremented the counter a


number of times equal to the smaller addend. As a sample size of 40 is not sufficient for structural modeling using latent variables, we modeled the relations between information processing parameter estimates with their SAT scores using hierarchical multiple regression analyses. Because several processes (e.g., MIN parameter estimates) showed nonlinear relations with age, we first forced AGE and AGE2 into the equation as predictors of an SAT variable, representing the linear and quadratic relations of age to the dependent variable. Then, two additional predictors were formed. For the first predictor, the parameter estimates for the MIN predictor and RMSE estimates were first standardized and then summed to form a predictor termed MINR. The MINR variable is an analog of the Addition Efficiency latent variable in the model presented in Figure 2, which used PRODUCT parameter estimates and RMSE as two indicators of a single latent variable. The MINR predictor thus represented addition efficiency for subjects using digital, counting processes to solve simple addition problems. The second additional predictor was a sum of the intercept and encoding parameter estimates, reflecting the basic Speediness latent variable in the model in Figure 2.

The results of these analyses for subjects using digital processes are presented in Table 4. As shown in Table 4, the MINR variable explained statistically significant amounts of variance in each of the three mathematics achievement subtests-MCOMP, MCONC, and MAPPL. Even after representing age trends in achievement, MINR led to AR2 values ranging from .13 to .19, all ps < .Ol. For the subtests that represented solely Verbal Comprehension or Verbal Achieve-

TABLE 4

Individual Difference Analyses with Information Processing Parameter Estimates And SAT Scores for Study 1: Subjects Using Digital Processes

Equation ‘-be R’

Model

R2

MCOMP = 1.83 (AGE) - 1.55 (AGE2) - .41 (MINR) MCONC = .44 (AGE) - .I6 (AGE2) ~ .42 (MINR)

MAPPL = -.40 (AGE) + .61 (AGE2) - .49 (MINR)

LCOMP = .72 (AGE) - .28 (AGE2) - .21 (MINR) LVOC = - .56 (AGE) + .89 (AGE2) - .25 (MINR)

RCOMP = 1.09 (AGE) - .91 (AGE2) - .48 (MINR)

RWSS = .75 (AGE) - .89 (AGE2) - .58 (MINR)

SPELL = 1.36 (AGE) - 1.22 (AGEZ) - .33 (MINR)

,265 ,395

,216 ,350

.I68 .356

,280 .314 .181 ,228

,187 ,363

,062 ,321

.I15 ,202

AR2 P

,130 c.01

,134 <.Ol

.188 <.Ol

.034 >.15 ,047 >.14

,176 <.Ol

,254 c.001

,087 c.06

No&: The tabled regression equations present standardized regression weights for the three predictors. AGE is the subject’s age

in years, AGE* is the square of AGE and represents the quadratic trend of age, and MINR represents the sum of standardized

estimates of the MIN parameter and the Root Mean Squared Error parameter from the MIN model tit to SubJects who were best predicted by MIN. The achievement variables arc: MCONC = Mathematics Concepts. MCOMP = Mathematics Computation. MAPPL = Mathematics Applications, LCOMP = Listening Comprehension, LVOC = Language Vocabulary, RCOMP =

Reading Comprehension, RWSS = Reading and Word study skills, SPELL = Spelling. Age RZ = the squared multiple COT-

relation with the linear and quadratic functions of age in the model, Model R * = the squared multiple correlation when the MINR variable was added into the preceding regression model, AR z = the difference between the two preceding R* values, and

p = the probability level of the F ratio, having I and 36 degrees of freedom, testing the significance of the hR*.

190 LEARNING AND lNDlVlDUAL DIFFERENCES WLIJME 4, NUMBER 3, 1992

ment-LCOMP and LVOC-neither MINR nor the speediness variable explained significant amounts of variance, ps > .14. But, for the subtests involving Reading Skills-RCOMP, RWSS, and SPELL-MINR provided AR2 values that were either statistically significant (ps < .Ol) or strong trends (p < .06). Impor- tantly, in all eight regression analyses, the sign of the regression weight for the MINR predictor was negative, indicating that the smaller the MINR score (i.e., the more highly automatized the counting process), the higher the level of achievement. Another important finding was that, after entering age trends and MINR into each equation, the speediness variable was never a significant predictor of achievement test scores, replicating a pattern of relations in the structural model for retrieval subjects presented in Figure 2.

DISCUSSION

The results of the present study have several important implications. The first implication involves the modeling of RT data at the group, subgroup, and individual levels. While virtually all extant research on addition using RT paradigms has based analyses and conclusions on group-level analyses of RT data, the results of the present study cast doubt on the propriety of this practice. Even at the second grade level, a majority of students appeared to use a memory retrieval strategy when responding to addition problems, yet a minority of sixth grade students still used counting strategies to solve addition problems. Thus, individual differences in strategy use appear to be the rule, rather than the exception, and research approaches should permit the modeling of such differences in strategy use for problem solution in order to portray accurately the development of skill in mental arithmetic.

Second, parameter estimates for different cognitive processes underlying mental addition show somewhat different age trends. The intercept and Search/Compute parameter estimates appeared to follow log-linear relations with grade, but the time for encoding of digits appeared to vary little with age. If these differential trends for different cognitive processes were replicated across samples and studies, such a finding would constitute a most compelling reason for pursuing the componential representation of the cognitive skills underlying mental arithmetic performance.

A third implication of the results of the present study concerns the external validity of information processing models of mental addition. For students using memory retrieval when responding to addition problems, the direct relations from the information processing latent variables to achievement latent variables revealed three major trends: Addition Efficiency was the only significant predictor of achievement latent variables, basic Speediness was not a predictor of achievement variables, and the Addition Efficiency latent variable explained rather more variance in Math Achievement than in other achievement latent


variables, but also explained a considerable amount of variance in a latent variable reflecting stored memory representations of words. This latter finding may imply the existence of a common retrieval component that activates items in long-term memory (cf. Horn 1988) and is consistent with results reported by Muth (1984), who found a moderate relationship (r2 = .37) between reading ability and computational skill. On the other hand, rather than forcing an interpretation on the relation between Addition Efficiency and Reading Skills, we presume that, if a cognitive component score reflecting word retrieval had been available on all subjects, the path from Addition Efficiency to Reading Skills would have been greatly attenuated.

For students apparently using counting processes to solve addition problems, regressions of SAT scores on age and processing parameter estimates also revealed strong and interpretable patterns among the achievement subtests, patterns that largely mirrored those for the retrieval subjects. Taken together, these results provide strong support for patterns of convergent and discriminant external validation among the information processing and achievement test measures.

STUDY 2

Most previous investigations of the development of cognitive components underlying addition performance (e.g., Ashcraft & Fierman 1982) have been rather limited in design; with few exceptions (e.g., Hamann & Ashcraft 1985), the cognitive component models considered have usually dealt only with simple addition. However, Widaman et al. (1989) proposed and tested a general model for mental addition that represents the cognitive components involved in solving addition problems of any size. The componential model provided for explicit estimation of parameter estimates reflecting time for (a) encoding digits, (b) retrieving or recomputing sums of two numbers, (c) carrying to the next column, (d) re-encoding digits in incorrect sums, and (e) response differences between correct and incorrect problems. Given the successful testing of the Widaman et al. model on a sample of college students, the present study was designed to extend the validation of the model to students in elementary school grades. The primary questions addressed (a) whether all components of the general model could be estimated reliably at several grade levels and (b) whether there were interpretable developmental trends in component process execution times as a function of grade. Two final aims of the present study were (c) to replicate developmental trends from Study 1 for model parameters estimated from RTs to simple addition problems and (d) to determine whether subgroup analyses would once again provide a more accurate representation of the development of skill in mental addition than would group-level analyses.

192 LEARNING AND lNDlVlDlJAL DIFFERENCES VOLUME 4. NUMBER 3, 1992

METHODS

SUBJECTS

A total of 163 students (79 males) were recruited for the study, 24 (12 males) in second grade, 20 (11 males) in fourth grade, 19 (11 males) in sixth grade, and 100 college undergraduates (45 males). The mean chronological ages for students were 7.4 years (SD = .60), 9.5 years (SD = .71), 11.4 years (SD = .67), and years (SD = 1.2) for students in second grade, fourth grade, sixth grade, college, respectively.

19.8 and

CHRONOMETRIC PROBLEM SETS

Simple Addition Problems. A representative sample of 40 of the 100 simple addition problems were selected for presentation. The major constraint invoked was that the digits 0 through 9 appeared an equal number of times as the first addend and an equal number of times as the second addend. No tie problems were used.

Complex Addition Problems. The complex addition problems used in the present study were ones in which one two-digit number was added to another two-digit number. There are 8100 possible “two-digit plus two-digit” complex addition problems, given the 90 two-digit numbers 10-99. A representative sample of 40 of the 8100 problems was selected, with the constraint that each of the digits O-9 appeared an equal number of times in each of the four positions (e.g., units column of the first addend) across the complex addition problems.

Combined Problem Set. Each of the 40 simple and 40 complex addition problems was presented once with the correct sum and once with an incorrect sum, resulting in a total of 80 simple and 80 complex addition problems. Incorrect sums differed from the correct sum by -tl or ?2 for the simple addition problems. For complex addition problems, half of the incorrect sums differed from the correct sum by ? 1 or *2, and the remaining half of the incorrect sums differed from the correct sum by ~10 (i.e., ?l in the tens column). The resulting set of 160 problems was presented to each subject on an Apple II microcomputer that ensured ? 1 ms accuracy of reaction times. The instructions, the stimuli, and performance feedback were presented as described in Study 1, and all temporal durations for warning signals and intertrial intervals were identical to those in Study 1.

PENCIL-AND-PAPER TEST BATTERY

For purposes of external validation of parameter estimates, three measures of Numerical Facility-Simple Addition, Complex Addition, and Simple Subtrac- tion-modeled after standard speeded tests (Ekstrom, French, & Harman 1976;


Thurstone & Thurstone 1941), and three measures of Perceptual Speed- Number Comparison, Finding A’s, and Identical Pictures-from the ETS battery of factor referenced tests (Ekstrom et al. 1976) were administered to each subject in the second, fourth, and sixth grade. To maintain comparability with analyses reported in Study 1, the external validation analyses reported here were based only on Study 2 elementary school subjects.

Due to a lack of time constraints during data collection with the college subjects, we administered a larger and different battery of tests to the college students. Geary and Widaman (1987) reported the external validation of model parameters in the college sample, revealing strong relations between model parameters related uniquely to mental addition (efficiency of retrieval and of carrying) and traditional paper-and-pencil measures of numerical facility.

Simple Addition. Simple addition problems were presented in columnwise format with a total of eight columns by nine rows or 72 problems per page. Each form consisted of two pages of randomly ordered problems with the first 100 problems of each being the basic 100 possible combinations of the digits O-9 over O-9. The remaining problems were randomly selected repeats from this population of problems.

Complex Addition. Complex addition problems were presented in columnwise format with a total of seven columns by eight rows or 56 problems per page. Each form consisted of two pages of two types of randomly ordered problems; these problems were selected from the population of possible combinations of two digits over two digits for half of the problems and the population of possible combinations of two digits over one digit for the other half. Problems were randomly ordered under the constraint that no more than three of any one problem type could occur in sequence.

Simple Subtraction. Simple subtraction problems were presented in columnwise format with a total of 80 problems per page. Each form consisted of two pages of randomly ordered problems selected from the population of 45 possible combinations of one digit over one digit when the subtrahends would produce a positive answer (i.e., the first subtrahend was always greater than the second).

Finding A’s. The two forms of the Finding A’s test from the ETS battery of factor referenced cognitive tests were presented (Ekstrom et al. 1976). Subjects were required to find the five words containing the letter “a” which were imbedded in a columnwise list of forty-one words consisting of five to nine letters each. Each form consisted of a total of 20 columns.

Identical Pictures. Also from the ETS battery (Ekstrom et al. 1976), the Identical Pictures test contained items that required subjects to choose which of five possible comparison figures or pictures best matched a standard figure. Each of the two forms consisted of 48 items.

194 LEARNING AND INDIVIDUAL DIFFERENCES WLUME 4. NUMBER 3. 1992

Number Comparison. The final ETS battery test, Number Comparison, consisted of two columns of pairs of random numbers. Subjects were asked to make same/different comparisons for each pair of random numbers by placing a “+” between the pairs of numbers which were the same and a “-” between the pairs of numbers which were different. The number strings ranged in length from 3 to 12 digits, with each pair having the same number of digits.

Procedure. During the first two days of testing, an experimenter entered each of six classrooms to conduct the group testing. Instructions for each test were given by reciting the written instructions and verbally reiterating the key points prior to each of the six individual tests. The complete testing session lasted approximately 45 minutes. For each of the six ability measures, two parallel forms were administered which were averaged into an overall score for each test; subjects were allowed 90 seconds or 120 seconds per form and were instructed to answer as many problems as possible in a specified order during the time allotted.

RESULTS

Analyses were performed separately for each grade level. For each grade, an average reaction time for each problem was computed across subjects and served as dependent variable. The procedures described in Study 1 for identify- ing outlier RTs were used again on the data from the present study. As will be described below, data from all 163 subjects to simple addition problems were used, but only 137 subjects provided useful RT data to complex addition problems. Using the procedures described in Study 1 to handle outliers, 405 RTs, or 3.11 percent, of the total of 13,040 RTs to simple addition problems and 296 RTs, or 2.70 percent, of the total of 10,960 RTs to complex addition problems were deleted. These rates of outliers were quite comparable to those reported in previous studies (e.g., Widaman et al. 1989). Finally, when modeling grade- related trends in parameter estimates, the college student sample was desig- nated as falling at the 14th grade level. The modal grade of the college students was 14 (i.e., sophomore, N = 76), and the average age of the college students was approximately 8 years greater than the age of the sixth grade students in the study, consistent with the difference in grade levels.

INTERNAL VALIDATION OF CHRONOMETRIC MODELS

Simple Addition: Group-Level Models. The analyses of mean RTs to the 80 simple addition problems at each grade level are presented in Table 5. The analyses revealed trends that were largely consistent with previous research (e.g., Ash- craft & Fierman 1982) as well as those reported in Study 1 (see Table 1). How- ever, in the present analyses, a memory network retrieval model provided the

DEVELOPMENT OF SKILL IN MENTAL ADDlTlON 195

TABLE 5

Summaries of Group Regression Analyses by Grade for Study 2: Simple Addition

Equation R2 F df RMSE

Grade 2 (IV = 24, i@ = 6251 ms) RT = 1822 + 541 (ENCODE) + 928.3 (MIN) ,828 185.77

+ 541 (REENC)

Partial F: 14.96, 311.14, 14.96

RT = 2333 + 496 (ENCODE) + 111 .O (PRODUCT) .831 190.07

+ 496 (REENC)

Partial F: 12.71, 318.57, 12.71

Grade 4 (N = 20, m = 3217 ms)

RT = 723 + 426 (ENCODE) + 349.6 (MIN) ,787 141.88

+ 426 (REENC)

Partial F: 40.49, 193.19, 40.49

RT = 916 + 404 (ENCODE) + 42.8 (PRODUCT) ,814 168.42

+ 404 (REENC)

Partial F: 41.46, 232.96, 41.46

RT = 604 + 322 (ENCODE)

+ 322 (REENC)

Partial F: 38.38, 148.37, 38.38

RT = 736 + 304 (ENCODE)

+ 304 (REENC)

Partial F: 41.63, 199.47, 41.63

RT = 591 + 100 (ENCODE)

+ 100 (REENC)

Partial F: 41.32, 132.80, 41.32

RT = 629 + 95 (ENCODE)

+ 95 (REENC)

Partial F: 44.04, 172.98, 44.04

Grade 6 (N = 19, E = 2424 ms)

+ 238.3 (MIN) .748 114.44

+ 29.8 (PRODUCT) ,795 149.12

College (N = 100, m = 1138 ms)

+ 67.3 (MIN) .736 107.53

+ 8.4 (PRODUCT) ,779 135.48

2,77

2,77

2,77

2,77

2.77

2.77

2,77

2-77

985.7

976.3

471.1

439.8

366.4

330.8

109.3

100.1

Now: All models significant at the p < .ooOl level. All partial F-ratios significant at the p < .Ol level

best fit to RT data at all four grade levels, although models having MIN and PRODUCT as structural variables fit the RT data from the second graders approximately equally well. Across grade levels, all parameters decreased systematically from second grade though college, with the intercept and Search/Compute stage parameters revealing the largest decreases between second and fourth grade and smaller changes thereafter, and encoding time estimates showing more linear decreases as a function of grade.

Simple Addition: Subgroup-Level Models. Because of the demonstration in Study 1 that group-level analyses may misrepresent the development of skill in mental addition, all analyses were performed on data from individual subjects. As

196 LEARNING AND INDIVIDUAL DIFFERENCES VOLUME 4, NUMBER 3, 1992

shown in Table 6, a total of 43 subjects were identified as using digital processes when responding to simple addition problems, 14, 4, 5, and 20 subjects at the second grade, fourth grade, sixth grade, and college levels, respectively. On the other hand, 120 subjects appeared to use retrieval when responding to the addition problems, consisting of 10, 16, 14, and 80 subjects at the second grade, fourth grade, sixth grade, and college levels, respectively. These results imply that a considerable minority of students, approximately 20 percent, at even rather advanced grade levels appeared to use digital, reconstructive processes when responding to addition problems, rather than using retrieval processes.

The results of constrained regression analyses for the different groups of subjects are also provided in Table 6. For both digital and retrieval subjects, the

TABLE 6

Summaries of Constrained Subgroup Regression Analyses by Grade for Study 2: Simple Addition

CJnrestricted Estimatim

Equrrtim R” RMSE

Digital Subjects (N = 43, E = 3336 ms)

Gra& 2 (N = 14, F? = 6281 ms)

RT = 1921 + 529 (ENCODE) + 919.1 (MIN) .X36 196.68

+ 529 (REENC)

Grudr 4 (A’ = 4, @ = 3521 ms)

RT = 496 + S29 (ENCODE) + 39X.2 (MIN) .776 133.40

+ 529 (REENC)

tirade 6 (N = 5. E = 2273 ms)

RT = 496 + 319 (ENCODE) + 248.4 (MIN) ,703 91.10

+ 319 (REENC)

College (N = 20, E = 1268 ma)

RT = 496 + 122 (ENCODE) + 107.9 (MIN) ,769 128.25 + 122 (REENC)

Network Retrieval Subjects (N = 120, E = 3233 ms)

Grade 2 (N = IO, z = 6210 ms)

RT = 2661 + 352 (ENCODE) + 119.9 (PRODUCT) ,783 139. IO

+ 352 (REENC)

Grade 4 (N = 16, E = 3141 ms)

RT = 999 + 352 (ENCODE) + 44. I (PRODUCT) ,779 133.40

+ 352 (REENC)

Grade 6 (N = 14, E = 2478 ms)

RT = 700 + 352 (ENCODE) + 25.9 (PRODUCT) .753 91.10 + 352 (REENC)

Cullqe (N = 80, FZ = II05 ma)

RT = 420 + 137 (ENCODE) + 8.9 (PRODUCT) .I42 128.25 + 137 (REENC)

Comtrairted

E.\timation

,836 .X36

776 ,775

.697 .69l

,758 ,741

,783 .783

,778 .77X

,747 .746

.688 .686

Nor?; All models significant at the p ( .oOOl level

DEVELOPMENTOFSKILL IN MENTAL ADDITION 197

constraints invoked had little effect on the fit of regression models to data, as evidenced by the small drops in the constrained estimates of X2 relative to the R2 estimates under unrestricted estimation. The results for digital subjects are shown in the top half of Table 6. The intercept had a nonlinear relation with grade, with a large value for second grade subjects and equality across the three later grade levels. Time for encoding digits in a problem and re-encoding digits in incorrect stated sums also exhibited a nonlinear change as a function of grade, with equality between the first two grade levels and log-linear change between fourth grade and college. Finally, replicating the trend from Study 1, parameter estimates for the MIN structural variable had a log-linear trend as a function of grade, with a large change between second and fourth grade and systematically smaller changes as grade increased.

The results of constrained regression analyses for retrieval subjects are presented in the bottom half of Table 6. For retrieval subjects, both the intercept and the PRODUCT parameter estimates followed log-linear relations with grade, exhibiting rather large decreases from the second to the fourth grade level, moderate decreases from the fourth to the sixth grade level, and relatively small changes to the college level. In contrast, digit encoding time was identical across the elementary school years and then rather lower for the college students.

Complex Addition: Group-Level Models. Largely because instruction in complex addition was just being initiated during second grade (California State Department of Education 1985), RTs to complex addition problems were collected only from students in fourth and later grades. In analyses of RTs to complex addition, we assumed that students solved such problems in a columnwise fashion, solving the units column first and then the tens column, a strategy that provided the best fit to data in the Widaman et al. (1989) study. To model this strategy, we defined separate structural variables to represent incrementing in the units and tens columns, structural variables termed UNITMIN and TENMIN, respectively. Parallel structural variables representing memory network retrieval were also specified, UNITPROD and TENPROD, respectively. When estimating parameters, we constrained the digital incrementing rate or the retrieval rate estimates to be identical during processing of the sums in the units and tens columns, a constraint supported by the analyses of Widaman and his associates (Widaman et al. 1989; Geary, Widaman, & Little 1986). As with simple addition problems, the times for encoding digits in the problem and for re-encoding digits in incorrect sums were also constrained to equality. A dichotomous variable, CARRY, was defined which indicated whether or not a carry from the units to tens column was required in the problem. Finally, we specified our regression models to reflect self-termination of processing if an error occurred in the units column of the stated sum; once again, this was supported by previous research findings (e.g., Widaman et al. 1989).

Analyses of RTs to complex addition problems revealed trends similar to those in earlier analyses, with systematically decreasing values of model parameters across the grades from fourth grade through college, as shown in Table 7. In

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0’8LI 9L‘f 0Z’f61 P88’

(ClOHdNXL) P’L + (AWIV3) P8Z +

(aONdlINn) P‘L + (XI03N3) 6SI + ZP6 = Dl

SL’98 ‘WLS ‘El ‘ZP ‘P9’LS ‘SL’98 1.4 IWJ~d (3NXItl) WI +

8’181 9L‘f

(NIKNXL) E.L~ + (mm3 lff +

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8’ff9 9L‘f (NIHNW S’SLZ + wRw3) S96 +

PZ’SOI 908 (NIW.LINfl) S’SLZ + (XI03N3) 6fI + LZZZ = .LH

(S=’ 99LP = LJ ‘81 = N) 9 JP”J3

60’81 ‘16’f6 ‘99’61 ‘16’f6 ‘60.81 :d le!pEd (3NXIH) EPZ +

S.089 9L’f (aOXdNXL) P’PE + (ArlD’3) 888 +

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(3NXIEI) 081 + (NIFINXL) L’XZE + (AtlW3) SLOI +

Z6’Pf I ZP8’ (NIKLINCI) L’SZE + (Fla03N3) 081 + PWZ = &I (=” SELS = 18 ‘61 = N) ti JP”J3 -

2661 ‘E H3fiWllN ‘P 3WfllOA 861


of subjects. Comparison of the levels of fit of the unrestricted and constrained models confirms that the constraints invoked had little substantial impact on the fit of the models to the data. As a result, only the estimates from the constrained models are presented.

The analyses of RTs to complex addition problems by the digital subgroups are shown in the top half of Table 8. Both the intercept and the time taken to carry showed comparable trends across grade levels, taking on identical values at the fourth and sixth grade levels and much reduced values in the college sample. The incrementing time parameter, the regression weights for the UNITMIN and TENMIN predictors, exhibited a nonlinear trend across grade, with a rather large change from Grade 4 to Grade 6 and then another substantial drop in the college sample. In contrast to the grade trends revealed by other parameters, the

TABLE 8

Summaries of Constrained Subgroup Regression Analyses by Grade for Study 2: Complex Addition

Equation

Unrestricted

Estimation

R2 RMSE

Constrained

Estimation

Ce = 0 Xe # 0

R2 R2

Digital Subjects

Grade 4 (N = 7, E = 5655 ms)

RT = 2064 + 160 (ENCODE) + 472.1 (UNITMIN) ,882

+ 879 (CARRY) + 472.1 (TENMIN)

+ 160 (REENC)

Grade 6 (IV = 9, ?ff = 4840 ms)


+ 879 (CARRY) + 305.1 (TENMIN)

+ 160 (REENC)

College (N = 27, if? = 2181 ms)


+ 140 (CARRY) + 67.6 (TENMIN)

+ 160 (REENC)

Network Retrieval Subjects

Grade 4 (N = 12, E = 5781 ms)

RT = 3277 + 179 (ENCODE) + 28.3 (UNITPROD) ,748

+ 1172 (CARRY) + 28.3 (TENPROD)

+ 179 (REENC)

Grade 6 (N = 9, m = 4693 ms)


+ 896 (CARRY) + 28.3 (TENPROD)

+ 179 (REENC)

College (N = 73, !@ = 2284 ms)


+ 275 (CARRY) + 7.3 (TENPROD)

+ 179 (REENC)

189.47 ,880 ,880

111.39 ,814 ,814

149.01 .844 .844

75.21 .748 .748

64.56 .718 .718

190.20 ,881 .881

Now: All models significant at the p < .OOOl level. All partial F-ratios significant at the p -C .Ol level

200 LEARNING AND lNDlV/DUAL DIFFERENCES VOLUME 4, NUMBER 3, 1992

speed of encodinglre-encoding digits was adequately modeled by an equality constraint across the three grade groups from fourth grade through college.

The results of constrained analyses of RTs to complex addition problems by retrieval subjects are presented in the bottom half of Table 8. As with the analyses of data from digital subjects, the intercept and carry parameters showed identical aging trends. In the present analyses, both the intercept and carry parameter estimates exhibited nonlinear trends as a function of grade, with a large decrease from fourth to sixth grade and then a somewhat larger drop between sixth grade and college. The retrieval time parameter, estimated as the regression weight for the UNITPROD and TENPROD structural variables, showed no change from fourth to sixth grade, but then a substantial decrease in the college sample. In contrast to the preceding trends, the encoding/re-encoding time parameter was best fit by an equality constraint across the three grade levels, replicating a pattern shown by the digital subjects.

EXTERNAL VALIDATION OF MODEL PARAMETERS

To demonstrate the external validity of model parameters, regression analyses similar to those performed on digital subjects in Study 1 were performed. First, two dependent variables were computed as follows: (a) the first dependent variable, NF, represented numerical facility and was obtained as the sum of standardized scores on the Simple Addition, Complex Addition, and Simple Subtraction tests, and (b) the second dependent variable, I’S, reflected perceptual speed and was obtained as the sum of standardized scores on the Finding A’s, Identical Pictures, and Number Comparison tests. As predictor variables, the linear and quadratic function of chronological age, AGE and AGE2, respectively, were first defined. Then, other predictors were formulated in the following manner: (a) for digital subjects, MINR was the sum of standardized values for the MIN parameter and the RMSE for each subject, (b) for retrieval subjects, PRODR was the sum of standardized values for the PROD parameter and the RMSE for each subject, and (c) for all subjects, the sum of parameter estimates for the intercept and encoding parameters was computed. Thus, MINR and PRODR represented efficiency of executing the cognitive processes central to the correct and speedy response to addition problems for digital and retrieval subjects, respectively, and the combined intercept/encoding time variable reflected basic speed processes. In specifying regression models, AGE and AGE2 were first forced into each equation to partial out age-related variance in the dependent variables. Then, MINR and the intercept/encoding speed variables were used as additional potential predictors for digital subjects, and PRODR and the intercept/encoding speed variables were used as potential predictors for retrieval subjects.

Parameters From Simple Addition. The external validation analyses for parameters estimated from simple addition problems are shown in the top half of Table 9. For digital subjects, MINR explained a statistically significant portion of NF


TABLE 9

Individual Difference Analyses with Information Processing Parameter Estimates And Ability Measures for Study 2, by Processing Strategy and Problem Type

Equation A@ R2

Model

R2 AR2 P

Digital Subjects, Simple Addition (N = 23)

NF” = .27 (AGE) + .45 (AGE2) - .27 (MINR) ,782 ,828 ,046 c.05

PSb = .68 (AGE) + .09 (AGE2) - .09 (MINR) ,684 ,690 .006 >.50

Network Retrieval Subjects, Simple Addition (N = 40)

NF = - .65 (AGE) + 1.22 (AGE2) - .33 (PRODR) ,621 ,614 .053 c.05

PS = .91 (AGE) - .28 (AGE2) -.I6 (PRODR) ,560 ,573 ,013 >.30

Digital Subjects, Complex Addition (N = 16)

NF = 5.38 (AGE) - 5.01 (AGE2) - .70 (MINR)c .407 ,841 ,434 c.001

PS = 4.87 (AGE) - 4.57 (AGE2) - .33 (MINR) .286 ,383 .097 >.I9

Network Retrieval Subjects, Complex Addition (N = 21)

NF = .16 (AGE) + .39 (AGE2) - .22 (PRODR) ,328 .372 .044 >.20

PS = 2.02 (AGE) - 1.54 (AGE2) - .28 (PRODR) .261 ,330 ,069 >.20

AWe: The tabled values are standardized regression weights for the three predictors. MINR ia the sum of standardized MIN and

RMSE parameters for digital subjects; PRODR is the sum of standardired PRODUCT and RMSE parameters for the retrieval

subjects. Age R2 = the squared multiple correlation with AGE and AGE2 in the model. Model RZ = the squared multiple

correlation when MINR or PRODR was added to the preceding equation, AR * = the difference between the two R*s, and

p = the statistical significance associated with the AR2. “Numerical Facility (NF) was the sum of standardized scores on the

Simple Addition, Complex Additon, and Simple Subtraction tests. bPerceptual Speed (PS) was the sum of standardized scores on

the Finding A’s, Identical Pictures. Number Comparison tests. ‘The combined intercept and encoding parameter also entered

into the equation with an increase in R2 = ,052. p < .04

variance (p < .05) over and above that explained by age trends, as hypothesized. In contrast, neither MINR nor the basic speed variable explained a significant amount of variance in the I’S dependent variable. A similar pattern held for the retrieval subjects, as PRODR explained a statistically significant amount of variance in NF (p < .05) after age trends were partialled, but neither PRODR nor basic speed explained substantial variance in PS.

Parameters From Complex Addition. For parameters estimated from complex addition problems, external validation analyses are presented in the bottom half of Table 9. For digital subjects, MINR explained a considerable amount of variance in the age-partialled NF variable (p < .OOl), and the basic speed predictor also explained a statistically significant amount of variance in NF. In contrast, neither MINR nor the basic speed variable explained significant amounts of variance in the age-partialled I’S variable. For retrieval subjects, there was a lack of evidence of significant external validity of parameter estimates as neither PRODR nor basic speed explained significant portions of age-partialled variance in either the NF or PS variables.

By way of summary, in three of the four sets of regression analyses, a predic-

202 LEARNING AND INDIVIDUAL DIFFERENCES WLUME 4, NUMBER 3. 1992

tor variable indexing efficiency of executing the cognitive processes unique to mental addition explained statistically significant portions of variance in a paper- and-pencil measure of numerical facility, but failed to explain significant variance in a paper-and-pencil measure of perceptual speed, exhibiting patterns of convergent and discriminant validation that were predicted on the basis of theory and previous research (e.g., Widaman et al. 1989; Geary & Widaman 1987). Moreover, in all analyses, the sign of the regression weight was in the appropri- ate, negative direction, indicating that the smaller the value on the predictor, and therefore the more efficiently the processes were performed, the higher the predicted value on the dependent variable reflecting numerical facility.

DISCUSSION

Study 2 was undertaken to replicate and extend previous findings with regard to developmental changes in efficiency in the cognitive processes underlying responding to simple addition and to extend the research by Widaman et al. (1989) on complex addition to younger age levels. The results of Study 2 provided fairly clear support for all of the research questions posed. First, for parameters based on RTs to simple addition problems, the intercept and Search/Compute stage parameter estimates for both digital and retrieval subjects exhibited rather nonlinear trends as a function of grade, with large decreases in parameter estimates from second to fourth grade and smaller decreases thereafter. In contrast, encoding time estimates typically showed little in the way of systematic developmental change for either digital or retrieval subjects. These differential developmental trends for the separate processing parameters replicated trends found in Study 1.

Second, parameter estimates for the cognitive processes underlying response to complex addition problems, based on the componential model proposed by Widaman et al. (1989), were estimated with adequate precision on data from most fourth and sixth grade subjects. Furthermore, for the students in fourth and sixth grades and in college, the estimates for the cognitive processes underlying complex addition revealed similar developmental trends as those based on simple addition. Specifically, the intercept and Search/Compute stage parameters revealed nonlinear patterns of change as a function of grade, whereas estimates of encoding time revealed little change across grade levels.

Replicating a finding from Study 1, the analysis of data from individual subjects and from subgroups of subjects, aggregated on the basis of common strategy use, provided a more accurate portrayal of developmental trends than did the analysis of data averaged across all subjects at a particular grade level. Developmental trends in parameter estimates from the subgroup-level analyses were more easily represented and interpreted, and the individual parameter estimates exhibited adequate patterns of external validity.

As a final note, the transition from use of digital, counting processes to re-

DEVELOPMENTOFSKlLL INMENTALADDlTlON 203

trieval processes when responding to addition problems was once again shown quite clearly in Study 2. A minority of students in second grade was identified as using retrieval when responding to simple addition problems, but approximately 80 percent of the students in fourth and sixth grades and in college appeared to use the more mature and efficient retrieval strategy.

DEVELOPMENT OF SKILL IN ARITHMETIC COMPUTATION AND RETRIEVAL: COMPARISONS ACROSS STUDIES 1 AND 2

There are at least two ways of comparing results across Studies 1 and 2 to characterize the development of the cognitive skills related to mental addition. One of these ways is to tabulate the numbers of subjects using digital and retrieval strategies across the two studies, a tabulation provided in Table 10. As shown in the top half of Table 10, the percentage of subjects using retrieval to solve addition problems rose from 57 to 64 and then to 81 across the second, fourth, and sixth grade levels, respectively, whereas the percentage of subjects using digital processes declined from 43 to 36 to 19 across the three grade levels, respectively. A similar trend in strategy use was shown in Study 2, results that are presented in the middle section of Table 10. In Study 2, 42 percent of the students in second grade used retrieval, whereas approximately 80 percent of

TABLE 10 Distribution of Subjects Using Digital and Retrieval

Stategies For Simple Addition, By Grade and Study

Grade

Total

N

Network

Digital Retrieval

N Pet N Pet

2

4

6

2

4

6

14

2

4

6

14

Study 1

37 16 43 21 51

45 16 36 29 64

41 8 19 33 81

Study 2

24 14 58 10 42

20 4 20 16 80

19 5 26 14 14

100 20 20 80 80

Studies I and 2 Combined

61 30 49 31 51

65 20 31 45 69

60 13 22 41 78

100 20 20 80 80


the students in the later grades used retrieval; contrasting trends were shown in the use of digital processes.

Combining these results across studies leads to the distributions presented in the bottom section of Table 10. Based on these data, there appears to be a rather smooth transition from digital to retrieval processes. That is, about 50 percent of second graders used retrieval, and this increased to approximately 70 percent in fourth grade, and then to about 80 percent in sixth grade and college. In comple- mentary fashion, the percentage of students using digital processes decreased from about 50 percent in second grade to about 30 percent in fourth grade and then to approximately 20 percent in sixth grade and college. These results imply that a substantial percentage of students in second grade are already using retrieval processes when responding to simple addition problems. Moreover, the results suggest that a sizeable minority of approximately 20 percent of students may never make a full transition to retrieval processes, relying instead on digital processes to solve addition problems.

A second way of comparing results from the two studies is to model the developmental changes in parameter estimates for the digital and retrieval subjects. As mentioned above, Kail (1986, 1988) administered a variety of cognitive processing tasks to samples of subjects varying in chronological age from 8 to 22 years of age. Kail then compared the fit of exponential and hyperbolic functions to the resulting cross-sectional estimates of cognitive processing parameters and found that the exponential function provided a somewhat better fit to the developmental data than did the hyperbolic function. The exponential function used by Kail has the following form:

PE = 11 + 6 ePcX,

where PE is the cognitive processing parameter estimate, a is an estimate of the asymptotic level of the PE, 6 is a multiplier used to determine the intercept of the curve, e is the natural logarithm, c is a decay parameter that governs how swiftly the asymptotic level of performance is attained, and X is the independent variable. In his analyses, Kail(l986, 1988) used chronological age as the independent variable, although other variables, such as grade in school, are reasonable alternative mediators of performance.

Given the nonlinear functions observed in analyses in Studies 1 and 2, we also fit two additional types of models to the developmental trends in parameter estimates. First, to demonstrate the need for a nonlinear model, we contrasted the fit of a traditional linear model to the fits attained by the nonlinear models considered. The linear model was of the form PE = a + 6X, where Q and 6 are the usual intercept and slope, respectively, in a standard linear regression equation for predicting PE from X. Second, we also fit a simple multiplicative function to the data, a function with the form:

PE = a X”.

DEVELOPMENTOFSKlLL IN MENTAL ADDITION 205

where a is a multiplier that determines the height of the curve, b is an exponent that governs the degree of curvature of the resulting function, and the other terms are as defined above. When fitting exponential, linear, and multiplicative functions to the developmental data, we performed all analyses twice, once using chronological age as the independent variable and once using grade in school as the independent variable.

In the cognitive processing model for simple addition presented in Figure 1 and in the more general model discussed by Widaman et al. (1989), the core component in models for mental addition is that representing the SearchKom- pute stage of processing. For digital subjects, this component corresponds to the MIN structural variable; for retrieval subjects, the PRODUCT structural variable represents this core component. In Figure 3, the MIN parameter estimates based on the unrestricted subgroup regression analyses are presented; the ten data points plotted represent three data points from Study 1 (grades 2,4, and 6), all of which were based on simple addition problems, four data points from Study 2 analyses of simple addition (grades 2, 4, 6, and college [14]), and three points from Study 2 analyses of complex addition (grades 4, 6, and 14). The MIN parameter estimates from the unrestricted regression analyses were used in

FIGURE 3

Plot of the relation behveen mean MIN parameter estimates and grade, combining results across Studies 1 and 2 for both simple and complex addition.

- MP4 = 2536 (GRADE)-’ 26

Observed Estimates:

l Simple, Study 1

0 Simple, Study 2

0 Complex, Study 2

? 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Grade

206 LEARNING AND lNDlVlDUAL DIFFERENCES VOLUME 4, NUMBER 3,1992

these analyses as a conservative procedure; if estimates from the constrained analyses had been used, the constraints invoked may have biased the estimates to have too uniform an appearance. The developmental trend in MIN parameter estimates was best fit by a multiplicative nonlinear equation using Grade as the independent variable, having the form:

MIN = 2536 * Grade - 1.26 I

which explained a large proportion of the variance among the estimates, X2 = .957, F(1,8) = 178.05, p < .OOOl. Given the exponent of -1.26 for Grade, which is close to - 1 .O, the MIN parameter estimates are roughly a function of the inverse of grade. The multiplicative function of Grade fit the data slightly better than did the multiplicative function of chronological age, X2 = .952, or the Kail (1986) exponential function, which had an R 2 = .955 whether age or grade was used as independent variable. In addition to its slightly better level of fit, the multiplicative function of Grade is more parsimonious than the Kail exponential function, as the multiplicative function has two parameters versus three parameters for the exponential function. Given these considerations, we selected the multiplicative function as providing the optimal description of the development of automaticity of counting speed by the digital subjects. The nonlinearity of the trend relating MIN estimates to Grade was also supported by the rather poorer fit of a linear slope to the data points in Figure 3, which resulted in R2 = .571.

In Figure 4, ten PRODUCT parameter estimates are presented that parallel the estimates presented in Figure 3 for the MIN structural variable. These parameter estimates were obtained from the unrestricted subgroup analyses based on retrieval subjects, including all estimates from Study 1 and from the analyses of simple and complex problems in Study 2. The relationship between age and efficiency of retrieval, indexed by the PRODUCT parameter estimates, was well described by the multiplicative nonlinear equation using Grade as independent variable, having the form:

PRODUCT = 200 * Grade - i.i6,

which explained a large proportion of the variance among the estimates, R2 = .795, F(l,B) = 31.08, p < .0006. As was the case in the analysis of MIN estimates, the exponent of Grade, -1.16, in this analysis was rather close to -1.0, suggesting that PRODUCT p arameter estimates vary approximately as a function of the inverse of grade. The multiplicative function of Grade fit the data better than did the multiplicative function of chronological age, X2 = .781. Also, despite having the smaller number of parameter estimates, the multiplicative function once again provided a somewhat better level of fit to the PRODUCT estimates than did the Kail (1986) exponential function, which had an R2 = ,781 whether age or grade served as independent variable. Not surprisingly, a linear regression model fit to the PRODUCT estimates in Figure 4 provided a rather poor repre-

DEVELOPMENTOFSKILL IN MENNTALADDITION 207

FIGURE 4

Plot of the relation between mean PRODUCT parameter estimates and grade, combining results across Studies 1 and 2 for both simple and complex addition.

120

110:

100:

90:

80 :

_^ /” :

60:

50:

40:

30 :

20 :

10 -

0 Redicred values:

- PRODUCT = 200 (GRADE) -I”’

Observed Estimates:

l Simple, Study 1

0 Simple, Study 2

0 Complex, Study 2

0 1 2 3 4 5 6 7 I3 9 10 11 12 13 14 15

Grade

sentation of the data, X2 = .504, indicating the clear superiority of the nonlinear trend relating PRODUCT estimates to Grade.

The graphs of the MIN and PRODUCT parameter estimates in Figures 3 and 4 revealed similar trends in the development of automaticity in the digital and retrieval processes, respectively, represented by the two variables. The major lack of fit of the nonlinear model occurred at the second grade level, where there was greater variability across studies in the parameter estimates for the MIN and PRODUCT structural variables. At later grade levels, parameter estimates derived from the two studies and across simple and complex addition data sets clustered fairly tightly at each grade level and fell close to the best fit multiplicative function.

That both MIN and PRODUCT parameter estimates were related approximately to the inverse of Grade suggests that Grade may provide a useful index of the amount of practice with or exposure to addition problems that occurs as a function of school experiences. Studies of the effects of practice on choice RT

have established that change in RT is a negatively accelerated function of trials (i.e., practice) on the task (Welford 1980). That is, there are relatively large reductions in choice RT during early stages of practice, with ever smaller reduc-

208 LEARNING AND lNDlVlDlJAL DIFFERENCES VOLUME 4, NUMBER 3. 1992

tions in choice RT as practice increases. Of course, changes in choice RT as a function of practice are rather microgenetic in nature (cf. Werner 1957), happen- ing over the course of practice that lasts a few minutes or a few days. In contrast, the changes in processing efficiency reflected in Figures 3 and 4 are rather macrogenetic, occurring as a function of years of maturation and schooling. Regardless of this difference in scope, Grade may be a most useful index of the amount of practice with or exposure to addition problems experienced by students during their years in school and may therefore be a powerful predictor of developmental changes in the efficiency with which students respond to simple and complex addition. The explanatory value of Grade also supports the sug- gestion by Horn and Cattell (1982) that systematic influences of acculturation, such as schooling, affect the development of such crystallized abilities as mental addition.

GENERAL DISCUSSION

In the present article, two studies were conducted to investigate the development of skill in mental addition. One of the more important outcomes from these studies was the analysis of RT data at the individual level and the important insights that this provided for the modeling of the development of skill in mental addition. In previous research (e.g., Ashcraft & Fierman 1982), regression modeling of RT data has usually been conducted on the average RT of all students at a particular grade level to each of the addition problems presented. Research of this type leads to conclusions regarding developmental shifts in strategy use that portray such shifts as occurring at approximately the same time for all individuals, for example, concluding that a shift is made during third or fourth grade from counting to retrieval processes (cf. Ashcraft 1982). However, the results of our studies revealed that the movement from digital, counting processes to retrieval of answers from a stored memory network is an individual difference variable. Even though a regression model with a digital structural variable (i.e., MIN) fit second grade group-level RT data better than did a regression model with a retrieval variable, approximately 50 percent of second grade students appeared already to use retrieval when responding to the simple addition problems. Moreover, the transition from digital to retrieval processes was not complete by fourth grade; a substantial minority of fourth graders, sixth graders, and even college students-30, 20, and 20 percent, respectively-appear to use digital processes and may never make the transition to primary reliance on retrieval of answers from a long-term memory network of addition facts. The results of the present studies suggest that the transition from use of a given strategy to a more mature one must be represented as an individual difference variable, in order to portray accurately the development of arithmetical skills.

A second major outcome was the estimation of the decidedly nonlinear pat-

DEVELOPMENT OFSKILL IN MENTAL ADDITION 209

terns of development in efficiency of the Search/Compute processes invoked during speedy response to addition problems. Thus, complementing the docu- mentation of the strategy shift from counting to retrieval processes, we found that mean MIN and PRODUCT parameter estimates exhibited a large decrease from second to fourth grade and showed systematically smaller decreases thereafter; similar trends have been reported by Kail (1986, 1988) for a variety of cognitive tasks, including mental addition. Our best fitting multiplicative models for MIN and PRODUCT parameter estimates indicated that these estimates were a negatively accelerated function of Grade, analogous to the effects on choice RT of practice over rather short periods of time, such as trials or days. As a result, Grade may be a useful proxy variable for the amount of practice children and adolescents have with arithmetic problems as a function of their years of schooling, consistent with hypotheses regarding crystallized abilities (Horn 1988; Horn & Cattell 1982).

The third major outcome of the present studies was the stable, consistent patterns of external validation of parameter estimates associated with the cognitive components identified in the model developed by Widaman et al. (1989). In Study 1, efficiency in executing the cognitive processes underlying response to addition problems was most strongly related to mathematics achievement scores and was also fairly strongly related to reading achievement for both digital and retrieval subjects. In Study 2, numerical processing efficiency was significantly related to a numerical facility measure and was not related to a measure of perceptual speed. Thus, across studies, parameter estimates based on the Widaman et al. componential model showed significant relations with achievement test scores and with ability test scores that were consistent with our a priori hypotheses. These results support the conclusion that the parameter estimates derived from the modeling of RT to simple and complex addition problems index cognitive skills that are related to or underlie achievement in school, demonstrating the potential diagnostic significance of the componential model for addition performance.

Although the present study supported the several hypotheses guiding the research, consideration of the overall pattern of results gives rise to at least two major issues that should be pursued in future research. First, longitudinal research should be conducted to determine whether data from the same individuals assessed at multiple points in time follow the same trends as those identified in Figures 3 and 4. Both of the studies in the present paper utilized cross-sectional designs, testing children of different ages at a single point in time and then attributing age or grade differences to the effects of maturation and schooling. To validate further our conclusions, longitudinal designs should be used to replicate the results of the present studies, to see whether intraindividual changes approximate the interindividual trends derived in the present study. A related point is the need to study the transition from counting to retrieval strategies when responding to addition problems. Attainment of a certain level of automatization of responding as a necessary condition for switch- ing from digital to retrieval processes is a tempting hypothesis to entertain.

210 LEARNING AND /ND/V/DUAL DIFFERENCES VOLUME 4, NUMBER 3. 1992

Research designed to identify the necessary and sufficient conditions for strategy shift to a more mature level would provide a major step forward for research in the arithmetic domain (cf. Siegler & Jenkins 1989).

Second, research should be conducted using collateral measures of strategy choice, such as obtained in research by Siegler and his associates (e.g., 1988a; 1988b; Siegler & Shrager 1984; Siegler & Taraban 1986), to determine the degree of generalizability of the present results. Using the Siegler approach, investigators gather trial-by-trial information on strategy use, for example, by observing the child as s/he solves problems for explicit strategy use or by questioning the child regarding the strategy invoked on each particular problem. Based on collateral measures, Siegler (1987; 1988a; 1988b) argued that children exhibit a great deal of variability in strategy use across arithmetic problems and that a single regression analysis of RTs to problems that are solved using different strategies may lead to rather biased, uninterpretable results. These criticisms by Siegler give rise to at least two considerations. The first consideration involves the relation between variability in strategy use and level of expertise. Variability in strategy use is likely to be negatively related to age or expertise on a task. In turn, this would attenuate the force of Siegler’s criticism for studies based on data from older children and adolescents, as in the present studies. That is, Siegler and his associates have typically used rather young children as subjects, aged 3 to 6 years of age, in studies on addition; subjects in the present studies tended to be rather older. The variability in strategy use reported by Siegler may be the rule in young children who are just learning how to add, and a failure to utilize collateral information when performing regression analyses of RT data may well lead to results that are difficult or impossible to interpret. However, variability in strategy use may be quite uncommon, or at least much more restricted, for older children and adolescents who have had much greater exposure to, and have greater facility with, addition problems (cf. Kaye 1986; Kaye, dewinstanley, Chen, & Bonnefil 1989). If this were the case, collateral information would have little influence on the modeling of RT data. This clearly is an issue that should be investigated empirically.

The second consideration concerns the potential biasing effects that may arise when obtaining the collateral measures of strategy use. That is, probing a child after each trial for strategy use information may lead to different patterns of strategy use than would occur under nonprobing conditions or may lead to inaccurate reports of the strategies used. In a recent study, Russo, Johnson, and Stephens (1989) investigated the effects of obtaining verbal protocols on performance on each of four types of task, including mental addition. In considering the potential effects of obtaining verbal protocols, Russo et al. distinguished between reactivity and veridicality of protocols. A verbal protocol would be considered reactive if obtaining the verbal protocol changed the primary cognitive processes underlying performance; in contrast, a verbal protocol would be nonveridical if it did not reflect the underlying cognitive processes accurately. Russo et al. found that there was evidence of reactivity with mental addition, as the obtaining of verbal protocols led to higher error rates when compared to a


“no protocol” control condition. Moreover, Russo et al. reported evidence of nonveridicality of verbal protocols across all tasks, including mental addition. Cooney, Ladd, and Abernathy (1989) also found evidence of reactivity in a study of the effects of obtaining verbal reports of strategy use on multiplication performance by third-grade students, finding that error rate was affected relative to a “no report” control condition. Given such findings, the effects of obtaining collateral strategy use information on arithmetic performance should be investigated further, to ensure that this procedure does not contaminate RT data even as it illuminates the types of strategies used across arithmetic problems.

Issues such as the preceding should be the focus of future research on cognitive processing of numerical problems, as each may lead to a need to qualify our conclusions based on the data from the present studies. On a more positive note, we trust that the results of the two studies in the present article will engender further research along these lines, seeking to develop a more adequate theory of the manner in which individuals develop skill in the domain of arithmetic.

ACKNOWLEDGMENTS: This study was supported in part by Grants HD-14688, HD-21056, and HD-22953 from the National Institute of Child Health and Human Devel- opment to the first author, an intramural grant from the Academic Senate, University of California at Riverside to the first author, a Humanities and Social Sciences research grant from the University of California at Riverside to the second author, and a grant from the Weldon Spring Foundation to the third author. Computing grants were provided through the Academic Senate and the Academic Computing Center at University of California at Riverside. The authors would like to thank the editors and an anonymous reviewer for their very helpful comments on earlier drafts of this article. Special appreciation is ex- tended to Mark Boyd, Roberta Cox, Mark Houser, Debbie Retter, and the faculty and staff of Longfellow Elementary School, Saint Paul Lutheran Elementary School, Seventh Day Adventist School, and the Riverside Unified School District for their assistance at various stages of the project.

REFERENCES

Ashcraft, M.H. (1982). “The development of mental arithmetic: A chronometric approach.“ Developmental Review, 2, 213-236.

Ashcraft, M.H. & J. Battaglia. (1978). “Cognitive arithmetic: Evidence for retrieval and decision processes in mental addition.” Journal of Experimental Psychology: Human Learning and Memory, 4, 527-538.

Ashcraft, M.H. & B.A. Fierman. (1982). “Mental addition in third, fourth, and sixth graders.” Journal of Experimental Child Psychology, 33, 216-234.

Ashcraft, M.H., B.A. Fierman, & R. Bartolotta. (1984). “The production and verification tasks in mental addition: An empirical comparison.” Developmental Review, 4, 157- 170.

Ashcraft, M.H. & E.H. Stazyk. (1981). “Mental addition: A test of three verification models.” Memory b Cognition, 9, 185-196.


Bentler, P.M. & D.G. Bonett. (1980). “Significance tests and goodness of fit in the analysis of covariance structures.” Psychological Bulletin, 88, 588-606.

California State Department of Education (1985). Mathematics framework for California pubhc schools: Kindergarten through grade twelve. Sacramento, CA: California State Depart- ment of Education.

Campbell, J.I.D. & D.J. Graham. (1985). “Mental multiplication skill: Structure, process, and acquisition.” Canadian Journal of Psychology, 39, 338-366.

Cattell, R.B. (1971). Abilities: Their structure, growth and action. Boston: Houghton-Mifflin. Cohen, J. & I’. Cohen. (1983). Applied multiple regressionlcorrelation analysis for the behavioral

sciences (2nd edition). Hillsdale, NJ: Erlbaum. Cooney, J.B., S.F. Ladd, & S. Abernathy. (1989, April). The influence of verbal protocol

methods on children’s mental computation. Paper presented at the meeting of the Ameri- can Educational Research Association, San Francisco.

Cooney, J.B., H.L. Swanson, & S.F. Ladd. (1988). “Acquisition of mental multiplication skill: Evidence for the transition between counting and retrieval strategies.” Cognition and Instruction, 5, 323-345.

Cornet, J., X. Seron, G. Deloche, & G. Lories. (1988). “Cognitive models of simple mental arithmetic: A critical review.” European Bulletin of Cognitizle Psychology, 8, 551-571.

Ekstrom, R.B., J.W. French, & H.H. Harman. (1976). Manual for kit of factor-referenced cognitive tests. Princeton, NJ: Educational Testing Service.

Geary, D.C. & K.F. Widaman. (1987). “Individual differences in cognitive arithmetic.” Journal of Experimental Psychology: General, 116, 154-171.

Geary, D.C., K.F. Widaman, & T.D. Little. (1986). “Cognitive addition and multiplication: Evidence for a single memory network.“ Memory & Cognition, 14, 478-487.

Groen, G.J. & J.M. Parkman. (1972). “A chronometric analysis of simple addition.” Psychological Review, 79, 329-343.

Hamann, M.S. & M.H. Ashcraft. (1985). “Simple and complex mental addition across development.” Journal of Experimental Child Psychology, 40, 49-72.

-. (1986). “Textbook presentations of the basic addition facts.” Cognition and Instruc- tion, 3, 173-192.

Horn, J.L. (1988). “Thinking about human abilities.” Pp. 645-685 in Handbook of multivari- ate experimental psycholpgy (2nd ed.), edited by J.R. Nesselroade & R.B. Cattell. New York: Plenum.

Horn, J.L. & R.B. Cattell. (1982). “Whimsey and misunderstandings of Gf-Gc theory: A comment on Guilford.” Psychological Bulletin, 92, 6233633.

Jdreskog, K.G. & D. Sorbom. (1988). LISREL 7: A guide to the program and applications. Chicago: SPSS.

Kail, R. (1986). “Sources of age differences in speed of processing.” Child Development, 57, 969-987.

-. (1988). “Developmental functions for speeds of cognitive processes.” journal of Experimental Child Psychology, 45, 339-364.

Kaye, D.B. (1986). “The development of mathematical cognition.” Cognitive Development, I, 157-170.

Kaye, D.B., I’. dewinstanley, Q. Chen, & V. Bonnefil. (1989). “Development of efficient arithmetic computation.” Journal of Educational Psychology, 81, 467-480.

Madden, R., E.F. Gardner, H.C. Rudman, B. Karlsen, & J.C. Merwin. (1974). Stanford Achievement Test (Intermediate Level II). New York: Harcourt Brace Jovanovich.

Miller, K., M. Perlmutter, & D. Keating. (1984). “Cognitive arithmetic: Comparison of operations.“ Journal of Experimental Psychology, 95, 437-444.

OEVELOPMENTOFSKILL IN MENTAL ADDITION 213

Muth, K.D. (1984). “Solving arithmetic word problems: Role of reading and computational skills.” Journal of EducationaI Psychology, 76, 205-210.

Parkman, J.M. & G.J. Groen. (1971). “Temporal aspects of simple addition and comparison.” Journal of Experimental Psychology, 89, 335-342.

Pellegrino, J.W. & S.R. Goldman. (1989). “Mental chronometry and individual differences in cognitive processes: Common pitfalls and their solutions.“ Learning and individual Differences, 1, 203-225.

Russo, J.E., E.J. Johnson, & D.L. Stephens. (1989). The validity of verbal protocols. Memory & Cognition, 17, 759-769.

Siegler, R.S. (1987). “The perils of averaging data over strategies: An example from children’s addition.” Journal of Experimental Psychology: General, 116, l-15.

-. (1988a). “Individual differences in strategy choices: Good students, not-so-good students, and perfectionists.” Child Development, 59, 833-851.

-. (198813). “Strategy choice procedures and the development of multiplication skill.” journal of Experimental Psychology: General, 117, 258-275.

Siegler, R.S. & E. Jenkins. (1989). How children discover new strategies. Hillsdale, NJ: Erlbaum.

Siegler, R.S. & J. Shrager. (1984). “Strategy choices in addition and subtraction: How do children know what to do?” Pp. 229-293 in Origins of cognitive skills, edited by C. Sophian. Hillsdale, NJ: Erlbaum.

Siegler, R.S. & R. Taraban. (1986). “Conditions of applicability of a strategy choice model.” Cognitive Development, 7, 31-51.

Thurstone, L.L. & T.G. Thurstone. (1941). “Factorial studies of intelligence.” Psychometric Monographs, No. 2.

Tucker, L.R. & C. Lewis. (1973). “A reliability coefficient for maximum likelihood factor analysis.” Psychometrika, 38, l-10.

Welford, A.T. (1980). “Choice reaction times: Basic concepts.” Pp. 73-128 in Reaction times, edited by A.T. Welford. New York: Academic.

Werner, H. (1957). “The concept of development from a comparative and organismic point of view.” In The concept of development: An issue in the study of human behavior, edited by D. Harris. Minneapolis: University of Minnesota Press.

Wheeler, L.R. (1939). “A comparative study of the difficulty of the 100 addition combinations.” Journal of Genetic Psychology, 54, 295-312.

Widaman, K.F., D.C. Geary, I’. Cormier, & T.D. Little. (1989). “A componential model for mental addition.” Journal of Experimental Psychology: Learning, Memory, and Cognition, 25, 898-919.

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