Modeling the mathematics achievement of Asian-American elementary students

MODELING THE MATHEMATICS ACHIEVEMENT OF ASIAN-AMERICAN ELEMENTARY STUDENTS PATRICIA A.WHANG AUBURNUNIVERSITY

GREGORYR.HANCOCK UNlVERSlTYOFMARYLAND,COLLEGEPARK

ABSTRACT: An examination is conducted of potential factors underlying the mathematics achievement of 60 fourth, 57 fifth, and 41 sixth grade Asian-American students attending public school in California. Specifically, structural equation modeling is used to evaluate hypothesized relationships among several constructs (Efficiency of Elementary Cognitive Processes, Working Memory Capacity, Reasoning Ability, Arithmetic Production, and Arithmetic Verification Speed), with particular interest directed toward these constructs’ ability to predict the Mathematics Achievement construct. The final structural model has an excellent fit to the observed variances and covariances and accounts for over three-fourths of the variability in the Mathematics Achievement construct (with Reasoning Ability being the largest explanatory construct included in the model). The predictive importance of all constructs within this generally high-achieving sample is discussed in the context of Ackerman’s (1988, 1989) theory of skill acquisition.

Widespread concern currently exists over the educational achievement of many students across the United States. From a simple economic perspective, in today’s world marketplace a less educated workforce is a less competitive workforce (Bishop 1989). Given society’s increasingly technological orientation, available jobs

Direct all corraspondence lo: Patricia A. Whang,4036 Haley Center, Auburn University, AL 36649-5221, Learning

and Individual

Dfflerences,

Volume

All rights of reproduction in any form reserved.

9, Number

1, 1997, pages 63-68.

Copyright

@ 1997

by JAI Press Inc. ISSN: 1041-6060

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will require higher skill levels in science and mathematics than in the past. A potential workforce without a fundamental understanding of these areas will precipitate repercussions felt not just by individuals alone. Our society as a whole will have to deal with increasing numbers of individuals prepared only for the decreasing numbers of nontechnical positions, as well as with its changing position in the world marketplace as a result (Rivera-Batiz 1992). Evidence that such concerns are not unfounded has surfaced in a multitude of cross-cultural comparisons that have consistently found our students lagging behind their Asian peers in mathematics achievement (Song & Ginsburg 1987; Stevenson & Lee 1990; Stevenson, Lee, & Stigler 1986). This concern was underscored in Time magazine, which reported that: It was a landmark study in 1980 that first raised U.S. consciousness about the math gap: elementary school students in both Japan and Taiwan rated far ahead of their American counterparts in mathematical skills. The shock-and an aftershock when a repeat survey in 1985 found the gap still there-galvanized parents, politicians and educators into placing a new emphasis on math and science in the schools (Time, January 11,1993, p.15).

Even though the public press had been raising the awareness and concern of the public at large, a 1990 follow-up study (Stevenson & Lee 1990) found that the disparity in performance still remained. In fact, the differences in performance appear to start as early as kindergarten and increase in magnitude as students get older (Stevenson et al. 1986). Several explanations have been offered. The successful performance of Japanese and Chinese students has been attributed to the centralization of educational policy (National Science Board 1983), the amount of time spent in school (Stigler, Lee, & Stevenson 1987), the amount of time spent on mathematics instruction (Mayer, Tajika, & Stanley 1991)‘ and the valuing of hard work over innate ability (Stevenson & Lee 1990). Interestingly, the international differences exposed in such studies are mirrored by intra-national differences within the United States. That is, Asian-American students, who are exposed to the same mathematics curriculum and methods of instruction, have been noted by several researchers to outperform their American peers of European and other ancestry groups (for a review see Chan & Vernon 1988; Flynn 1991). Despite a growing interest in Asian-American achievement, however, Sue and Okazaki (1990) point out that our understanding of these students’ achievement is fraught with uncertainty. Some have suggested that a more complete understanding will not be attained until we have more explicit information on the loci of performance differences within particular subject matter areas (Geary, Fan, & Bow-Thomas 1992). For example, interest in mathematics achievement is shifting to tasks that allow for the direct measurement of factors underlying the development of numerical skills. Given the increasing importance of mathematics in today’s society, and the relative success of Asian-American students in this area, a closer examination of Asian-American mathematics achievement seems warranted. The present study attempts to provide such an examination, proposing a structural model that draws from the cognitive psychology literature (as seen in Figure 1). As will be expanded upon, the following

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MODELING MATH ACHIEVEMENT

FIGURE 1 Hypothesized structural model.

constructs are hypothesized to have direct and/or indirect effects on mathematics achievement: elementary cognitive processing (ECP) efficiency, working memory (WM) capacity, reasoning ability, and knowledge of basic arithmetic facts. The theoretical basis for this proposed model is presented below.

ELEMENTARY COGNITIVE PROCESSING EFFICIENCY In the proposed model the construct of ECP Efficiency is included based on the assumption that differences in basic biologically-determined parameters of information processing ultimately influence the development of complex intellectual skills and knowledge (Anderson 1983; Brody 1992; Carroll 1980; Jensen 1992a). This fundamental construct is believed to be manifested in two key facets, the speed and the consistency of processing; operationally, ECP Efficiency is a secondorder construct born of the covariation between two first-order constructs, ECP Speed and ECP Consistency. ECP Efficiency is modeled in this manner because elementary cognitive processes (ECPs) are believed to be so basic (i.e., only engage a few processes) that the tasks designed to tap them are capable of being understood and performed correctly by most people; hence, only speed of response

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reliably differentiates performance (Jensen 1987), and for this reason experimental psychologists studying reaction time (RT) have generally been uninterested in intra-individual variability. However, with specific regard to IQ in&a-individual variability has been found to be a more reliable correlate than RT (Jensen 1992b). Thus, in the current study both the speed of responses and the consistency of the speed of responses will form the basis by which ECP Efficiency is judged. Speed of information processing, which is indicated by one‘s RT to various tasks, derives its theoretical importance from the fact that it is thought to reflect how information is transmitted from one cortical region to another via axons. Quickness in traversing and crossing synapses to linking dendrites is thought to affect information processing at every level of cognitive complexity, including the resultant response, in a beneficial manner because of the capacity limitations of the information processing system (Jensen 1991). More simply, minimizing the time necessary to process information should increase the availability of resources for application to other aspects of the task or to new tasks. As noted previously, the degree of consistency of RT across a number of trials has been found to correlate even more highly with complex cognitive behaviors than RT (Jensen 1987; 1992b). Intra-individual variability, as measured by RT standard deviation (RTsd), is hypothesized to reflect the amount of “noise” or errors in the transmission of information in the nervous system (Eysenck 1987). Jensen (1991) explains as follows: Neural oscillation acts as “noise” in the nervous system that degrades the efficiency of information processing. A rapid rate of oscillation is more favorable to g than a slower rate. This can be explained in terms of a simple analogy. If we think of oscillation as a “neuronal shutter,” analogous to the shutter of a camera, then the more rapid and shorter the duration of the “open” and “shut“ phases of the shutter, the less will be the moment-to-moment detail that is lost, or shut out, from a continuous input of stimuli and the chaining of operations while processing information in WM (p. 140).

While ECP Efficiency is hypothesized to have an effect on Reasoning Ability, as correlations have supported in so many studies (see Vernon 1987), it is the investigation of the potential effect on Mathematics Achievement that has largely eluded prior research. In one of very few studies, Spiegel and Bryant (1978) found correlations of -0.4 between mean RT and measures of mathematics achievement. When controlling for IQ however, the correlations became practically nonexistent. It should be noted that some of the tasks used by Spiegel and Bryant were considerably more cognitively complex than the processes of interest in the present study. More specifically, their matrix analysis task, included as a measure of ECPs, was similar to Raven’s Progressive Matrices, a commonly used measure of g. The present focus is on much more elementary processes that presumably underlie general intelligence as well as mathematical ability. Furthermore, measures of consistency were not included as part of Spiegel and Bryant’s study. I-Ience, there is still much to be learned about the importance of these early information processing steps to mathematics achievement.

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WORKING MEMORY CAPACITY As shown in Figure 1, ECP Efficiency is believed to have a direct impact on WM Capacity. The rationale for this hypothesized path derives from the fact that WM, which is responsible for the concomitant storage and processing of information (Baddeley 1986), is a limited capacity and rapid decay system. In order for information to be successfully stored or processed, its activation must exceed some minimum threshold; if an inadequate amount of activation is available to the system (i.e., less than the amount required to perform the task), then some of the activation maintaining old elements will be reallocated, producing a loss of information (Just & Carpenter 1992). Therefore, a key to WM Capacity is the ability to perform ECPs efficiently-the more mental operations that can be performed per unit of time before information decays in WM or before the system is overloaded, the greater the WM Capacity. This interpretation is consistent with the notion that the faster the flow of information in the processing system, the greater the functional capacity of WM (Lehrl & Fischer 1988). As is also relevant to the hypothesized model, several studies have supported the hypothesized link between WM Capacity and Mathematics Achievement (e.g., Hiebert, Carpenter, & Moser 1982; Kaye, DeWinstanley, Chen, & Bonnefil1989); in fact, many of the studies have explored this link specifically with exceptional populations (e.g., Dark & Benbow 1990,199l; Geary, Brown, & Samaranayake 1991). Geary et al. (1991), for example, posited that relatively poor WM resources of mathematically challenged students may contribute to their inability to form adequate representations of basic arithmetic facts, which necessitates continued reliance on effort-laden strategies such as counting. In short, considerable support exists for the hypothesized relationship between WM Capacity and Mathematics Achievement.

REASONING ABILITY Both ECP Efficiency and WM Capacity are hypothesized to be determinants of Reasoning Ability, which is typically “considered to be at or near the core of what is ordinarily meant by intelligence” (Carroll 1989, p. 56). Because of the constraints of our information processing system, individual differences in how efficiently the processing is performed are hypothesized to relate to individual differences in the performance of the complex behaviors in which they result. Therefore, it should not be surprising that virtually every elementary cognitive task has been found to correlate significantly with psychometrically measured general intelligence. Certainly when elementary cognitive tasks are very simple (e.g., simple and choice RT tasks) and when sufficient practice has been accorded, their correlations with general intelligence are quite low (Kyllonen 1985; Vernon 1987); nonetheless, generally high positive correlations have been found between WM capacity and reasoning ability (Kyllonen & Christal 1990). One interpretation given for why reasoning ability may reflect differences in WM capacity is that solving reasoning problems requires intensive use of attentional resources. Thus, the availability of resources is believed to increase the likelihood that encoding, continued activation of attributes and relations, or regulation of performance can take place.

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The importance of reasoning ability to the performance of complex behaviors is well established. For example, individual differences on measures of g have been shown to be better predictors of performance in school or college, military training programs, and business or industry than any other single psychological variable or combination of variables independent of g (Hunter 1986; Jensen 1984; Thorndike 1986). While there are certainly other predictors of success as well (see Sternberg 1996; p. 13), reasoning ability has been found to indirectly effect work sample performance, in that “people with higher mental ability acquire more job knowledge, and job knowledge, in turn, is the major determinant of work sample performance (Schmidt & Hunter 1992; p. 90). In factor analytic work, mathematics tests typically load heavily on what is identified as a g factor, with correlations ranging from 0.55 to 0.70 (Krutetskii 1976). In fact, it has been suggested that high 8 is the single most important factor for success in mathematics (Wrigley 1958). PRODUCTION

AND VERIFICATION

OF BASIC ARITHMETIC

FACTS

With respect to the arithmetic precursors of Mathematics Achievement, two are used in the current study: how quickly basic arithmetic problems are verified (Arithmetic Verification Speed) and how many can be answered in a given time period (Arithmetic Production). The reason for including both constructs in the current study is that both types of tasks have been used to investigate the nature of numerical cognition (see Ashcraft 1992). In production tasks, a problem is presented (e.g., 8+7=_) and the correct answer is the desired response. Verification tasks, in contrast, provide both the problem and an “answer” (e.g., 8+7=13), where the object of this task is to determine the verity of the stated answer. These two types of tasks are thought to provide potentially different insights into numerical cognition because the presence of an answer in verification problems may alter the solution strategy (e.g., working backwards), and possibly the retrieval outcome (Ashcraft 1992). In fact, Zbrodoff and Logan (1990) have empirically demonstrated that verification of response is probably more than fact production plus answer comparison. Instead, they suggested that verification could involve evaluating the amount of activation generated by an equation against some pre-set criterion level. Most recently, Campbell and Tarling (1996) have provided further evidence that the memory processes used for solving arithmetic production tasks may be significantly different than those used for verification tasks. As proposed in the model, Reasoning Ability and both ECP Efficiency and WM Capacity are believed to have direct causal effects upon Arithmetic Verification Speed and Arithmetic Production Speed. Quite simply, this is because inefficient processing or inadequate capacity would be expected to result in information being lost or degraded, which in turn may affect the integrity of the information being processed and hence the resulting response. In fact, as early as the second month of kindergarten, differences in the types of strategies used have been found to be related to the number of addition problems these young students were able to solve (Geary, Bow-Thomas, Fan, & Siegler 1993). These differences may be precipitated by early experience with numbers and arithmetic that result in the formation of associations between problems and answers (Siegler 1986; Siegler &

69

Shrager 1984). For example, to solve the problem 3 + 4, children initially rely on counting. The answer generated by counting, preferably 7 in this case, appears to become associated with the stated problem. Counting skills, in fact, appear to be related to WM Capacity and to the speed or articulating number words (Geary et al. 1993). Thus, initial reliance on counting to solve a problem eventually results in the direct retrieval of the number most strongly associated with the problem (Siegler 1986). Utilizing precious resources to perform laborious strategies for tasks that many do automatically is problematic, because fewer resources are available for coordinating and performing other aspects of the task or for attaining a conceptual understanding of the work being carried out. Because of the hypothesized influence of WM Capacity, ECP Efficiency, and Reasoning Ability on both Arithmetic Production and Arithmetic Verification Speed, the common causal antecedents should facilitate covariation among the two arithmetic constructs. In addition to this covariation, however, many mathematics-specific reasons must exist for these two constructs to covary other than those three common causal antecedents. For this reason, the model in Figure 1 depicts an additional covariance relationship between Arithmetic Production and Arithmetic Verification Speed. Specifically, the two constructs’ disturbance terms (representing construct variability not accounted for by the constructs‘ causal antecedents) were hypothesized to covary in this initial model. Allowing disturbance terms to covary, rather than allowing their constructs to covary directly, is necessary because the covariance of two endogenous constructs such as these is never a parameter to be determined in structural equation modeling. Allowing endogenous constructs’ disturbance terms to covary, as done in this case, indicates an additional (although unmeasured) hypothetical source of covariation between the constructs beyond those directly modeled. Finally, the proposed model posits the possible existence of direct effects of basic arithmetic fact knowledge (Arithmetic Production and Arithmetic Verification Speed), and its causal precursors, on Mathematics Achievement. As theorized by Kaye (1986) in his model of how mathematical proficiency develops, some critical level of computational efficiency may be required for the acquisition of higher level conceptual and procedural skills in mathematics. In fact, Geary and Burlingham-Dubree (1989) g arnered support for Kaye’s hypothesis, finding that mental addition’s strategic and speed-of-processing parameters were strongly related to skill at solving verbally presented word problems. In addition, further experimentation led to the conclusion that performance of basic arithmetic problems is predictive of skill acquisition in more complex mathematical domains (Geary et al. 1991). For example, automatic fact production may maximize the resources available for more sophisticated mathematical problem solving by decreasing the demand placed on the limited resources of WM (Hitch, Cundick, flaughey, Pugh, & Wright 1987). These conclusions are also consistent with crosscultural research by Geary et al. (1992), who suggested that the advantage held by the Chinese children over American children in solving addition problems seemed to reside in both the developmental maturity of the mix of strategies used to solve the problems and the response speed to simple addition problems. Thus, the retrieval (i.e., production and verification) speed of basic arithmetic facts may

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be important components for understanding mathematics achievement, as hypothesized in the current model. In sum, as seen in Figure 1, the proposed model starts with the more basic cognitive attributes of ECP Efficiency and WM Capacity. These are hypothesized to be influential in the determination of individuals’ Reasoning Ability, all of which may have an effect on their facility with fundamental arithmetic fact retrieval (Production and Verification). Ail of these elements, then, may in turn have an ultimate causal bearing on individuals’ Mathematics Achievement. The extent to which this hypothesized model fits the data, and the process of examining that fit, are described next.

METHOD SUBJECTS The school from which participants were recruited was a predominantly Chinese-American San Francisco Bay Area public school. Participants selected for this study were 158 Asian-American fourth, fifth, and sixth grade students (77 males and 81 females), of whom 115 reported speaking mainly Chinese at home, 21 Vietnamese, and 22 English. English was the primary language spoken in school, though most teachers or their classroom aides could speak Chinese. The total mathematics performance on the Comprehensive Test of Basic Skills (CTBS) for the fourth, fifth, and sixth graders selected for this study averaged at the 99th percentile for all three groups. All students selected for this study had returned signed parent permission slips, and were offered a pencil and sticker for their participation. Student ages ranged from 8 to 13 years (M=30.44; SD=l.O3).

CONS~UCTS AND MEASURED VARIABLES Nineteen measured variables were used in the modeling process, serving as indicators of seven first-order constructs. A description of all constructs, indicator variables, and the covariate follows, and in many cases includes explanations of the measures’ administrative procedures.

ECP Speed. This construct is believed to represent a general measure of the speed with which fundamental processes requisite to more cognitively complex behaviors are performed. The indicators of this construct were three chronometric tasks using an apparatus similar to that described by Jensen and Munro (1979). Participants were instructed to perform each task as fast as possible without making errors. They were given as many practice trials as desired before beginning testing. The measures obtained were as follows: Simple Reaction Time median (SRTm)---Participants keep a “home” button depressed until another single button illuminates. They then release the home

MODELlNGMAlHACHIEVEMENl

71

button as quickly as possible and move to depress the lighted button. SRTm is the median time in milliseconds (across 20 trials) between the target light illuminating and the release of the home button. Choice Reaction Time median (CRTm)--Participants keep a “home” button depressed until another single button randomly chosen from a group of eight possible buttons illuminates. They then release the home button as quickly as possible and move to depress the lighted button. CRTm is the median time in milliseconds (across 32 rials) between the target light illuminating and the release of the home button. This measure relates to Hicks law (1952), stating that reaction time increases linearly as a function of the logarithm of the number of choice alternatives. Oddman Reaction Time median (ODDRTm)-Participants keep a “home“ button depressed until three buttons from a group of eight possible buttons illuminate. They then release the home button as quickly as possible and move to depress the lighted button farthest from the other two. ODDRTm is the median time in milliseconds (across 36 trials) between the target lights illuminating and the release of the home button. This Oddman Paradigm is intended to measure speed of spatial discrimination (Frearson & Eysenck 1986).

ECP Consistency. This construct is intended to measure the degree to which participants’ reaction times are consistent across trials. Greatly varying reaction times are indicative of inconsistency and inefficiency in the functioning of elementary cognitive processes. The measures serving as indicators of this construct derive from the chronometric tasks described previously. Specifically, for each participant’s reaction times across the block of trials a standard deviation (sd) measure was computed. Thus, three measures existed: SRTsd, CRTsd, and ODDRTsd, standard deviations across trials for the respective chronometric tasks. ECP Efficiency. As described previously this represents a second-order construct with ECP Speed and ECP Consistency as its first-order indicators. That is, ECP Efficiency is derived from the covariation between ECP Speed and ECP Consistency. Conceptually, it is believed that someone having a high degree of ECP Efficiency will, as a result, be more expedient and more consistent in their expediency than someone having less ECP Efficiency. WM Capacity. Figuratively size of participants’

speaking, this construct is meant to assess the relative workbenches. Four measured variables serve as its indicators.

W~odc5ck-~ohnsun-revised, Memory for Words (Word Span, WS)--Standard inst~ctions were used to administer this test. Participants were required to repeat lists of unrelated words in the correct sequence. The words were presented at a rate of one per second until three consecutive strings, which ranged from one to eight words, were recalled incorrectly. The maximum number achieved successfully served as a participant’s measure.

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WISC-R, Digit Span (DS)--Standard instructions were used to administer this test. Participants listened to a series of digits recited orally by the examiner and then repeated them back from memory. There were two series of digits for each sequence length, which ranged from three to nine digits. Testing discontinued when both trials at any string length were failed. The maximum number achieved successfully served as a participant’s measure. WISC-R, Backward Digit Span (BDS)-The standard instructions required participants to repeat a string of digits in reverse order. Similar to the DS measure, testing was discontinued when the participant failed both trials at any string length, which in his case ranged from two to eight digits. The maximum number achieved successfully served as a participant’s measure. Add-a-Digit (AAD)-This measure was devised as an additional measure of WM capacity. As with other measures of WM capacity, successful task performance requires the simultaneous retention and manipulation of information. In this measure participants were read a string of digits; their task was to repeat back the digits after adding, as cued by the researcher, either one or two to all numbers in the string. Thus, when presented with the string “4-7-2,” the correct response would be “5-S-3, ” when cued to add one to each number. There were two series of digits for each sequence length, which ranged from two to six digits; the task was discontinued after three consecutive errors. The maximum number achieved successfully served as a participant’s measure.

Reasoning Ability. This construct was indicated by a single measure, Raven’s (1938) Standard Progressive Matrices. This is a nonverbal test of reasoning believed to be one of the best measures of psychometric general intelligence (Jensen 1987). This measure was group administered to each participating class using standard directions. Forty-five minutes was allotted to complete the test and all children were able to finish within this time frame. Arithmetic Production. This construct is meant to represent the facility with which one is able to retrieve basic arithmetic facts. The indicators for this construct consist of scores on single-digit multiplication (MUL), addition (ADD), and subtraction (SUB) tests. A three-page color coded packet containing one page of 100 addition problems (all possible single-digit problems), one page of 55 subtraction problems (all possible single-digit problems in which the subtrahend does not exceed the minuend), and one page of 100 multiplication problems (all possible single-digit problems) was group administered in classes. Participants were allowed 45 seconds per page, and were told to solve as many of the problems on each page as they could. The researcher started and stopped the test and insured that students were not working ahead by checking to see that all students were on the same colored page.

Arithmetic Verification Speed. This construct represents the speed with which participants

can determine

the verity

of simple

arithmetic

equations.

The indicator

MODELING MATH ACHIEVEMENT

73

variables for this construct came from a computerized mathematical verification test, in which single-digit addition, subtraction, and multiplication equations were presented. Briefly, participants keep a “home” button depressed until a singledigit equation appears on a computer screen. As quickly as possible the participant judges whether the equation is correct (e.g., 2+4=6) or incorrect (e.g., 9-3=5), then releases the home button to depress a “right” button or “wrong” button. Addition, subtraction, and multiplication verification reaction times (ADDVRT, SUBVRT, and MULVRT, respectively) are each calculated as the median time in milliseconds (across 20 trials) between an equation’s appearance and the release of the home button. Any equation to which a participant responded incorrectly was cycled to the end of the queue and presented again until a correct response occurred. Only measures from correct responses were used. Participants were given 15 practice trials (five in each area) prior to the task.

Mathematics Achievement. This construct was assessed by two indicators, both based on scores from the Comprehensive Test of Basic Skills (CTBS) found in participants’ school records. This group test is administered to students in California public schools over several days each spring, with strict time limits for each subtest. The mathematical portions of the CTBS are designed to measure how well students have acquired skills required for effective use of number in everyday living and for further academic study. The first mathematics subtest used, concepts and applications (CONC), assesses the areas of numeration, number sentences, number theory, problem solving, measurement, and geometry. The second subtest used, computation (COMP), assesses multiplication and division of whole numbers, and the addition, subtraction and multiplication of decimals and fractions. Age Covariate. Any variability

in the above measured variables possibly due to developmental differences (i.e., age-related differences) was partialed from the observed variables prior to modeling. That is, prior to fitting the observed covariante matrix with the hypothesized model shown in Figure 1, variability in all measured variables shared with age (measured in months) was removed. Thus, the findings of this study are not confounded by potential linear effects of age on the variables involved in the modeling process.

GENERALPROCEDURE All tasks other than the CTBS were administered in two separate sessions, conducted in separate classrooms. For the first session the Standard Progressive Matrices was administered first, followed by the Arithmetic Production measures. An hour was allocated for this session. During the second session measures of WM capacity and the computerized tasks were individually administered by one of four testers. One of the testers spoke Chinese in the infrequent event that participants with limited English ability required clarification of task instructions. Half of the participants completed the memory tasks first, while the other half did the computerized tasks first. Approximately an hour and a half was allocated for the second session.

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OVERVIEW OF ANALYSES The analysis of relations among the factors of interest proceeded in two general phases (see Anderson & Gerbing 1988; James, Mulaik, & Brett 1982), measurement and structural. Specifically, in the first phase a model in which first-order factors were allowed to covary was imposed upon the variance-covariance matrix (with age partialed out); this creates a model where any existing poorness of fit is necessarily contained in the measurement model relating observed variables to latent constructs. As is common practice, the measurement model was evaluated in this phase to see if any meaningful improvements could be made. Such improvements may take the form of dropping unnecessary indicators, adding cross-loadings (where a single observed variable serves as an indicator for more than one construct), or allowing pairs of residuals to covary. In the current study the only measurement model modification made was of the last type. The decision to allow a covariance between two residuals was made only if four criteria were met: (1) its inclusion would make theoretical sense, (2) it would make a significant improvement in the model, (3) the magnitude of its contribution would be substantial enough to be certain that overfitting (i.e., capitalization on chance covariation) is not occurring (see Byrne 1994), and (4) it would not yield offending estimates anywhere in the model. As will be presented in the results, the covariance between only one pair of residuals met these criteria; it was added to the measurement model. Before proceeding to the structural phase of analysis, a second type of measurement model was examined in this study. Given that our hypothesized structural model incorporates a second-order factor (ECP Efficiency), a confirmatory factor model was imposed upon the data in which the second-order factor was allowed to covary with all other non-ECP first-order factors. The point of this intermediate analysis was two-fold. First, doing so allows the derivation of a correlation matrix describing the relationships of factors among which the structural relations were hypothesized. Second, given that the hypothesized structural model is directly nested within this second-order confirmatory factor model, it serves as a baseline against which to compare the structural model statistically. In the second major phase of analysis, the initial structural model shown previously in Figure 1 was imposed upon the respecified measurement model. That is, the covariance paths among latent constructs in the second-order confirmatory factor model were replaced with the theoretically predetermined paths. After fitting this structural model to the data, its specification was investigated to see if any empirically meaningful improvements could be made. In general, such improvements may involve adding or removing structural paths. In the current study no paths were added; only those paths not making a significant contribution were removed from the model. The final structural model was then fit to the data. All analyses were performed using EQS (Bentler 1989), choosing maximum likelihood as the method of estimation. This method has been shown to be fairly robust to potential violations of multivariate normality (e.g., Muthen & Kaplan 1985), and for that reason is often recommended (e.g., Hayduk 1987). Also, one adjustment was made in the model in order to conform to proper modeling procedures. Because Reasoning Ability has only the Raven score as its indicator, it was necessary to fix the

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error variance to a function of l-.86, where .86 is the reported reliability of the instrument (Raven, Court, & Raven 1983). Finally, it should be noted that the sample size used in the current study is relatively small for modeling the proposed structural relations; as a result, power analysis of several key results is presented.

RESULTS DESCRIPTIVE STATISTICS The zero-order correlations among age and the 19 variables used in modeling appear in the Appendix above the matrix diagonal. The correlations among the 19 variables with age partialed out, the covariances among which were used in modeling, appear below the diagonal. Also included in the Appendix are the means of all measured variables for fourth, fifth, and sixth graders. In addition, as described previously, a non-causal model was imposed upon the covariance matrix in which the second-order factor ECP Efficiency and the firstorder factors of WM Capacity, Reasoning Ability, Arithmetic Production, Arithmetic Verification Speed, and Mathematics Achievement were all allowed to covary (the fit of this model is described below). The correlations among these constructs (with age partialed out) are presented in Table 1. Note that the firstorder constructs of ECP Speed and ECP Consistency are omitted because they define the second-order construct ECP Efficiency, which is in turn related to other first-order constructs in subsequent modeling.

MODEL FITTING In the first phase of analysis the first-order confirmatory factor analysis model was imposed upon the covariance matrix with age partialed out. The fit of this model is described in Table 2, as is the fit of all subsequent models. The chi-square value

Correlations I

TABLE 1 Among Constructs 2

3

4

1.

ECP Efficiency

2.

WM Capacity

.221*

3.

Reasoning

Ability

.310*

.315*

4.

Arithmetic

Production

.318*

.517*

.464*

5.

Arithmetic

Verification

.361*

,043

,139

.371*

6.

Mathematics

.324*

.432*

.814*

.690*

Note:

*p < .os

5

1.000

Speed

Achievement

1.000 l.OCG 1.000 1.000 .328*

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would seem to indicate that the model fits poorly; this is not surprising, however, given the well-known sensitivity of chi-square tests to sample size. While many alternative indices of fit have been proposed and reviewed (e.g., Gerbing & Anderson 1993; Tanaka 1993), one of those currently recommended and offered by EQS is the comparative fit index, or CFI (Bentler 1990). CFI values ranging from .90 to 1.00 have generally been regarded as reflecting acceptable fit of a model, although given substantial statistical and theoretical evidence models at the lower end of this range may receive minor respecification to achieve more satisfactory overall fit (see, e.g., example of post hoc model fitting in Chapter 3 of Byrne 1994). While the CFI is above .90 in the current case, the initial model was examined for potential improvement through statistically and theoretically meaningful respecification. For the current model such respecification took place using information from the multivariate Lagrange multiplier modification indices offered by EQS, resulting in the addition of one covariance path between residuals as per the criteria described previously. This covariance was between the residuals of ODDRTm and ODDRTsd, possibly an artifact of the ODDRT method. While allowing other pairs of residuals to covary would have yielded further significant improvement in the overall fit of the model, the modification indices estimated the chi-square contribution of the one pair to be substantially above the others (56.617, followed by 9.339, 8.575, etc.). Choosing only the first pair in this case, as per Byrne’s (1994, Chapter 3, Post Hoc Model Fitting) discussion, is more likely to represent meaningful respecification rather than overfitting. As seen in Table 2 the model chi-square value decreased. This is expected given that in the initial hypothesized model all inter-residual covariances were effectively constrained at zero, while in the current model one of those constraints was freed. Given that a hierarchical relationship exists between the two models, the difference between chi-square values follows a chi-square distribution with the number of degrees of freedom equal to the difference in degrees of freedom between the two models. Specifically, X2diff= 64.256 with dfdif= 1; this represents a significant (p<.OOl) improvement in the measurement portion of the model. The improvement in fit is also reflected in the value of the CFI, which indicates an excellent fit of the modified model (CFI=.974). This measurement model respecification was carried through all subsequent modeling.

TABLE 2 Summary of Model Fit Statistics Model

First-order

confirmatory

First-order

DFA model with covarying

df

p-V&e

234.158

132

<.OOl

,932

169.902

131

,013

,974

2

factor analysis (CFA) model residual pair

CFI

Second-order

CFA model*

175.160

136

.013

,913

Hypothesized

structural

175.160

136

,013

,973

184.706

136

,013

,972

Final Structural Note:

* G/h

model*

model*

covrrr~ing

residurrl

pair

As mentioned previously, an intermediate second-order confirmatory factor analysis model was imposed upon the data where the second-order ECP Efficiency construct was allowed to covary with all other non-ECP constructs. This model fit extremely well (CFI=.973), and serves as a reasonable baseline against which to compare the subsequent structural models. In the structural modeling phase the hypothesized structural model was implanted within the second-order confirmatory factor model. The fit of this model was identical to that of the second-order confirmatory factor model because all possible paths existed among the relevant constructs. Further, this model contained a number of structural paths that were not statistically significant. For this reason the model was refined using the multivariate Wald test offered by EQS, which identifies structural paths whose removal would not significantly decrease the overall fit of the model. This approach was taken rather than simply deleting nonsignificant paths because the deletion of one structural path may affect the magnitude, and hence significance, of other paths within the model. The multivariate Wald test estimates the simultaneous contributions of paths to the model, identifying those that

FIGURE 2 Final structural model.

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FIGURE 3 Final measurement model.

do not collectively contribute to the fit of the structural model. Seven such paths were identified and removed from the model (all of which were, in fact, nonsignificant in the previous model), the fit of which is summarized in Table 2. Notice that its CFI stilf indicates exceIlent fit. Furthermore, a hierarchical comparison of this final structural model and the previous model indicates a nonsignificant (p-.216) decrement in overall fit from the previous model (x2dw=9.544, +&~=7). This final model and its associated measurement model (without the covarying pair of residuals represented) appear in Figures 2 and 3, respectively. All paths, shown in their standardized metric, are statistically significant at the .05 level

MODELING MATH ACHIEVEMENT

79

One alteration in the presentation of the final structural and measurement models has been made to clarify interpretation. Conceptually, with most constructs a greater amount of that construct is represented by a larger number. Greater speed, however, is indicated by smaller numbers. Similarly, greater consistency as summarized by standard deviation measures is represented by smaller numbers. This fact has a direct bearing on the signs (+ or -) of the paths related to ECP Efficiency, which is defined by ECP Speed and ECP Consistency, and on those related to Arithmetic Verification Speed. However, rather than have the potential for confusion regarding the sign of these relationships, the signs of the indicator variables have been reflected for the purpose of defining all constructs in the same way. Specifically, by having all measured indicators for ECP Speed, ECP Consistency, and Arithmetic Verification Speed load negatively,, as shown in Figure 3, all of these constructs as well as the second-order construct of ECP Efficiency may be interpreted such that larger numbers conceptually represent greater amounts of speed, consistency, and efficiency. The interconstruct correlations previously shown in Table 1, as well as all remaining discussions, are presented with this reflection assumed.

INDIRECTEFFECTS As represented in Figure 2, the links between a pair of constructs may be many and varied. At the simplest level, one construct may have a hypothesized direct effect on another. ECP Efficiency, for example, is believed to have a direct effect on Reasoning Ability. Indirect effects, on the other hand, represent the hypothesized causal effect of one construct on another as mediated by one or more intervening constructs. WM Capacity has no direct effect on Mathematics Achievement, but it is hypothesized to influence Reasoning Ability, which in turn is hypothesized to cause a change in Mathematics Achievement. Whether or not this apparent indirect effect of WM Capacity on Mathematics Achievement may be assumed to be a non-chance one, however, is a matter for statistical tests to decide. In the final model shown in Figure 2, five indirect effects are possible: (1) ECP Efficiency on Arithmetic Production (through Reasoning Ability); (2) WM Capacity on Arithmetic Production (through Reasoning Ability); (3) Reasoning Ability on Mathematics Achievement (through Arithmetic Production); (4) ECP Efficiency on Mathematics Achievement (through Reasoning Ability, and through both Reasoning Ability and Arithmetic Production); and (5) WM Capacity on Mathematics Achievement (through Arithmetic Production, through Reasoning Ability, and through both). Of these five possibilities, EQS showed four to represent statistically significant (pc.05) indirect effects, only that of WM Capacity on Arithmetic Production was not significant. Of particular interest for the present study is the significance of the indirect effects on Mathematics Achievement. In addition to the strong direct effect of Reasoning Ability shown in Figure 2, it has an indirect effect on Mathematics Achievement as well. Thus, its standardized total effect on Mathematics Achievement is .748 (i.e., .618 + .309x.420). Furthermore, although neither WM Capacity nor ECP Efficiency

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have direct effects on Mathematics Achievement, both have significant indirect effects. These indirect effects are mediated by both Arithmetic Production and Reasoning Ability. In fact, all but one significant indirect effect on Mathematics Achievement involves Reasoning Ability. Thus, for explaining Mathematics Achievement, Reasoning Ability is a significant direct contributor, a significant indirect contributor, and a major mediator through which both ECP Efficiency and WM Capacity are believed to be influential.

DISCUSSION In the present study we sought to provide a preliminary description of the cognitive factors related to mathematics achievement for Asian-Americans, and those factors’ inter-relationships. The final structural model had an excellent fit to the observed variances and covariances, provided a much more parsimonious representation of the relationships of interest than originally hypothesized, and accounted for over three-fourths (i.e., 1-.4742) of the variability in the Mathematics Achievement construct. Several points with regard to this final structural model are worthy of elaboration. First, the path from ECP Efficiency to WM Capacity did not prove to represent a statistically significant direct effect. This path was originally posited based upon the documented belief that efficiently processing information would minimize the limitations of WM, namely the rapid decay of information, storage limitations, and storage-processing trade-offs. As Jensen (1993) discusses, when the capacity of WM is exceeded processing is thought to break down and, consequently, potentially important information may be lost. A formulaic explication of the relationship between ECP Efficiency and WM Capacity is even hypothesized by Lehrl and Fischer (1988). They state: C bits = S bits/ set x D set, where WM Capacity (C, measured in bits of information) is the product of the speed of information flow (S, measured in bits per second) and the duration time of information in WM when unrehearsed (0, measured in seconds). Given that such a relationship does in fact exist, there may be several reasons why our findings did not support its existence. Among them is the fact that we did not include a measure of duration, which according to Lehrl and Fischer’s formula is important to understanding WM Capacity. The inclusion of such a measure could potentially alter the present finding. In addition, the lack of statistical significance may be due to restriction of range in the scores from tasks constituting the WM Capacity construct. And, as always, the sample size may have afforded too little power to detect the potentially small effect of ECP Efficiency on WM Capacity; the standardized path coefficient prior to deletion was .198, with p=.O80. Following a strategy outlined by Saris and Satorra (1993) for power analysis without specific

MODELING MATH ACHIEVEMENT

81

path value alternatives (i.e., using the sample-based path estimate), a post-hoc estimate of the power for detecting significance (p < .05) of this particular path was found to be .44. This implies that if the standardized path in the population had actually been the small value of .198, the current study would have only had a 44% chance of detecting it. With regard to the outcome construct of interest, it is noteworthy that Mathematics Achievement was directly affected by Arithmetic Production, but not by Arithmetic Verification Speed. This finding supports Ashcraft’s (1992) suggestion that production and verification tasks may provide different insights into numerical cognition. Other than weak statistical power to detect Arithmetic Verification Speed’s potentially small effect (the standardized path coefficient prior to deletion was .126, with p=.103; power estimated at .37), a possible reason for this result may lie in the nature of solution strategies for production and verification tasks and their relation to the types of tasks typically required on group-administered standardized achievement tests. Specifically, most traditional tests such as the CTBS are in a multiple-choice or other selection-type format. As has been discussed previously (e.g., Hancock, Thiede, Sax, & Michael 1993), solution strategies in selection formats may include any or all of the following: solving for the answer or retrieving it from memory based only upon the multiple-choice stem (i.e., production); ruling out clearly incorrect options (i.e., verification); or working backwards from several options to converge upon the correct answer (i.e., verification). Thus, either or both production and verification strategies may be operating in selection-type tests such as the CTBS. Given the absence of a relationship between Arithmetic Verification Speed and Mathematics Achievement, however, it would appear that the ability to produce a response is a more valuable precursor to Mathematics Achievement. Another key result was the statistical non-significance of paths from both ECP Efficiency and WM Capacity to Mathematics Achievement. Although power estimates are low (.06 and .lO, respectively), as with all low post-hoc power estimates this may be the result of very small or non-existent effects the standardized path coefficient for ECP Efficiency prior to deletion was .017 (p=.840) and for WM Capacity prior to deletion it was .065 (p=.512). This is not to say, however, that these constructs have no bearing upon Mathematics Achievement; rather, their effects are indirect ones, largely mediated by Reasoning Ability. Individuals’ levels of ECP Efficiency and WM Capacity are significant determinants of their Reasoning Ability, as has been posited by past researchers (Jensen 1993; Kyllonen & Christal 1990; Vernon 1987). Reasoning Ability in turn, then, is an explanatory precursor of Mathematics Achievement. In fact, according to the final model, it is the largest such explanatory construct, which is consistent with Wrigley’s (1958) finding of a significant correlation between mathematical ability and general intelligence in an all White sample of elementary students. In the current study, though, the magnitude of the predictive value of Reasoning Ability is especially impressive given the somewhat restricted range of individual differences in mathematics achievement for the high achieving group under examination. The importance of Reasoning Ability when trying to understand Asian-American mathematics achievement is particularly interesting in the context of Ackerman’s

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(1988, 1989) theory of individual differences in skill acquisition. According to Ackerman, skill acquisition is divided into three continuous phases. In the first or cognitive phase, initial performance of a skill is constrained by mental ability and WM capacity. As an adequate cognitive representation of the task is formed, performance becomes more automatic (Logan 1988). During the second phase, automatic processing begins to speed performance, to enable parallel processing and reduce cognitive effort, and to enable new strategies to develop and be utilized to achieve performance (Fisk & Schneider 1984). Increasing experience with a task results in the learning of specific solutions to specific problems, which are retrievable during subsequent encounters with the problem. In other words, a knowledge base is being compiled. Because this information is assumed to be stored permanently in long-term memory, the demands on WM are reduced which should lead to more expedient and more accurate problem solving. During this phase, performance is also limited by a construct Ackerman calls perceptual speed ability, which is derived from test intercorrelations and clusterings much the same way as g (Ackerman 1989). This construct manifests itself in individual differences in the speed with which simple cognitive test items can be easily completed; in the current study, then, perceptual speed would be evidenced in performance on the Arithmetic Verification and Arithmetic Production tasks. Finally, in the third or autonomous phase, because the task has become automatized only psychomotor abilities are thought to constrain performance. That is, automatized task performance depends more on non-cognitive abilities than on cognitive abilities because little or no new information is being processed. In the current study, the ECP tasks used were very similar to those employed by Ackerman (1988) as measures of psychomotor ability. Thus, performance of a skill during the third phase of skill acquisition is thought to be constrained by psychomotor ability, or in the context of the current investigation, by the ECP Efficiency. Consistent practice, then, results in performance that is predominantly a function of automatic processing (Ackerman 1988, 1989; Logan 1988). Hence, during the third phase the skill is carried out in a fast and virtually effortless manner, theoretically constrained only by motor limitations. Transitioning from algorithmic processing to memory-based processing has been attributed to an accumulation of separate episodic traces with experience (Logan 1988). This notion is very similar to Siegler and Shrager’s (1984) Distribution of Associations model, in that memories become stronger because each experience lays down a separate trace that may be recruited at the time of retrieval, based on the strength of the available traces. Given Ackerman’s proposed model, the finding that Reasoning Ability is playing such a central role in understanding the mathematics achievement of Asian-Americans suggests that many of these students may still be in the first phase of skill acquisition, where performance is slow and laborious because processes have not become automatized. The parameters of performance are thus highly constrained by individuals’ ability to reason in the face of nonautomatized tasks, that is, their general mental ability. Had participants been in the second stage of skill acquisition, factors such as ECP Efficiency and WM Capacity would not have been expected to play as prominent a role because

83

MODELING MATH ACHIEVEMENT

increased algorithmic efficiency and memory retrieval would be more heavily relied upon to solve the more familiar aspects of the tasks at hand. Corroborating this interpretation is the fact that an arithmetic production measure, and not verification, is contributing directly to predicting mathematics achievement. Thus, many aspects of the tasks with which students are faced may not have been consistently practiced, and hence the finding that these constructs, in particular Reasoning Ability, are important to understanding the mathematics achievement of Asian-Americans.

CONCLUSION The current study has examined the mathematics performance of AsianAmerican elementary students and some of the hypothetical antecedents of that performance. We focused solely on Asian-Americans because our intent was to try to develop a better understanding of this group’s performance. Understanding how a system operates is necessary before making comparisons with other more or less efficient systems, for without such an understanding performance differences between systems may naively be attributed to any trait differences in those systems regardless of their actual relevance to the systems’ efficiency. The current study has sought such a systemic understanding through the development of a structural model. We fully recognize that this model is incomplete; because it is impossible to account for all potential causal elements in a system one could argue philosophically that any structural model is misspecified to some extent. However, the proposed model is meant to be a glimpse into a system, and it is a glimpse guided by extensive theory and one that has great success accounting for mathematics achievement in the Asian-American elementary students examined. The next logical step, then, should involve gathering data to facilitate intergroup comparisons. Such comparisons, when armed with a better understanding of a general system underlying mathematics performance, are necessary to identify sources of group performance differences and, most importantly, to facilitate meeting the educational needs of all American students (Asian-American and otherwise). It is these next comparative and ultimately remedial steps that we urge researchers to begin taking.

This study was supported by the Institute for the Study of EduACKNOWLEDGMENT: cational Differences. The authors wish to thank Gabriel Paulson, Natalie Ghan, and Robin Henke for their assistance in the data collection.

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APPENDIX ZERO-ORDERCORRELATlONS(ABOVEDIAGONAL),CORRELATlONSWlTH AGEPARTlALEDOUT(BELOWDIAGONAL),ANDVARlABLEMEANSBYGRADE I

2

3

4

5

6

7

8

9

IO -

,260

,382 ,362

,306 ,205 ,500

-.080 -.097 -.I38 -.I20

-a42 -.032 -.049 -.030 ,386

-.I65 -.097 -.I31 -.094 .346 .42 I

-.I03 -.046 -.I03 -.083 .420 ,438 ,416

-.080 -.Ol5 -.093 -.035 ,412 .472 .497 .728

-.074 -.077 -.I43 -.040 ,285 ,273 .72 I ,469 .5X4

I. ws 2. BDS 3. DS 4. AAD 5. SRTsd 6. CRTsd 7. ODDRTsd 8. SRTm 9. CRTm IO. ODDRTm I 1.RAVEN 12. ADD 13. SUB 14. MUL 15. ADDVRT 16. SUBVRT 17.MULVRT 18. COMP 19. CONC

,272 ,385 ,318 -.087 -.055 -.I80 -.I06 -.I03 -.098 ,139 ,298 ,269 ,272 -.023 ,027 -.03 1 .230 .I75

,366 ,235 -.I 10 -.058 -.I34 -.054 -.068 -.I29 ,252 ,381 ,294 ,345 -.060 -.054 -.074 ,239 ,285

,502 -.I42 -.057 -.I42 -.I06 -.I07 -.I55 ,141 ,254 ,193 ,174 -.087 -.076 -.087 ,165 ,048

-.I33 -.057 -.I32 -.09 I -.089 -.097 ,192 ,403 .247 ,323 -.08 I -.089 -.I I4 ,185 ,070

t?leclnS 4th, n=60 5th, n=57 6th, n=4 I

5.00 5.04 5.27

4.08 4.18 4.75

5.83 6.02 6.07

3.98 4.32 4.67

I 18.63 102.33 117.80

332.87 255.58 288.57

239.90 181.18 196.62

389.40 361.67 382.62

931.03 871.02 857.10

754.10 671.04 649.82

SDS (n=l58)

0.70

I.17

I.17

I .02

79.65

159.31

89.89

75.40

135.45

175.82

II

12

I3

14

15

16

17

I8

19

AGE

,132 ,241 ,136 I 80 -.2lO -.241 -.206 -.I59 -. 147 -.I54

,289 ,336 ,254 ,357 -.I86 -.076 -.263 -.280 -.224 -.218 ,325

,257 ,264 I85 ,214 -.I80 -.094 -.313 -. 102 -.I00 -.I74 ,297 ,618

,265 ,293 I66 ,260 -.068 -.I I3 -.I97 -.I59 -.I I8 -.I58 ,371 ,706 ,659

,238 .258 ,171 ,204 -.200 -.I79 -.233 -.I05 -.229 -.18l .617 ,448 ,429 .47 1 -.300

,203 ,345 ,064 ,122 -.035 -.I 13 -.I25 -.098 -.I27 -.085 ,521 .327 .347 ,449 -.I94

1. ws 2. BDS 3. DS 4. AAD 5. SRTsd 6. CRTsd 7. ODDRTsd 8. SRTm 9. CRTm 10. ODDRTm 1 I. RAVEN 12. ADD 13. SUB 14. MUL 15. ADDVRT

,326 ,307 ,357 -.I25

,633 ,777 -.373

,656 -.28 I

-.420

,392 ,354 .422 ,417 ,295 -.216 -.202 -.I93 -.I01 ,083 .09x ,084 -.I93 -.Ol5

.007

.m -.07l -020 Ix1 .I I3 320 .w7 2.54 384 -.I04 -.27 I -.23 I -.314

,437 ,439 ,488 ,299 -.250 -.I32 -.I21 -.I70 .I.50 I 84 .I79 -.I67 -.076

,061 ,009 -058 -025 076 ,148 ,325 .I16 .2?xJ .435 -.I22 -.275 -252 -307 ,897

,416 ,526 ,736 -.219 -.326 -.343 -.277 ,363 ,369 .37 1 -.212 -.069

.an -.a% -.069 -0% ,059 .I39 324 .I35 237 .4oM -.I IS -.298 -.240 -3% ,862

.7l I ,461 -.I62 -.269 -.I09 -.I59 ,107 .I25 .I42 -.I01 -.085

,619 -.I65 -.32 I -.I53 -.248 ,320 ,363 ,313 -.199-.053

-.I71 -.32l -.224 -.286 ,440 ,489 ,469 ,152 -.Ol I

,094 ,192 ,063 ,200 -.08 I -.I40 -.213 -.046 -.283 -.294 ,085 ,465 .2l I .540 -.312

(W&i,lU&)

MODELING MATH ACHIEVEMfNT

85

APPENDIX (continued) II

-i2

IJ

14

16.S’JWVRT

-.143

17. MULVRT 18. COMP 19. CONC

-.137

-.382 -.413 ,363 .174

-.302 -.294 ,403 ,280

-.421 -.466 .3s7 ,242

menns 4th, ~60 5th, n=57 6th, n=41

.607 ,485

39.32 42.8 1 43.15

27.19 34.15 41 .?S

8.32

It.26

28.92 32.65 34.25

22.31 30.52 38.95

9.09

11.01

15

I6

,907 ,876 -262 -.107 1450.9 1065.9 949.3

17

,891 ,901 -.234 -ml 1390.2 1032.4 940.7

-.I87 -040 1528.8 1089.2 953.9

I8

19

-.273 -.228

-.182 -.136 ,576

,575 86.10 84.88 87.37

81.62

20.31

22.47

75.89 74.68

AGE -.327 -.359 -.070 -.233 29.53 10.52 I 1.55

SDS (n= 158)

547.12

522.69

600.75

1.01

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