

Effects of Early Acceleration of Students in Mathematics on Attitude and Anxiety Toward Mathematics: The Developmental Perspectiveby Xin Ma  2003 This study examined the effects of early acceleration of students in mathematics on the development of their attitudes toward mathematics and mathematics anxiety across junior and senior high school. Data were derived from the Longitudinal Study of American Youth (LSAY). Hierarchical linear modeling (HLM) analyses showed that attitudes declined in the same degree between accelerated and nonaccelerated gifted and honors students, but declined significantly faster in accelerated than nonaccelerated regular students. Accelerated gifted students did not increase their anxiety. Anxiety grew at a similar rate between accelerated and nonaccelerated honors students, but accelerated regular students increased their anxiety significantly faster than nonaccelerated regular students. Once students were accelerated, most variation in rates of attitude and anxiety change was at the student rather than school level. Racial/ethnic background was the most important factor influencing rate of change at the student level. School contextual characteristics were major factors influencing rate of change at the school level. This study examined the effects of early acceleration of students in mathematics on the development of their attitudes toward mathematics and mathematics anxiety across junior and senior high school. Data were derived from the Longitudinal Study of American Youth (LSAY). Hierarchical linear modeling (HLM) analyses showed that attitudes declined in the same degree between accelerated and nonaccelerated gifted and honors students, but declined significantly faster in accelerated than nonaccelerated regular students. Accelerated gifted students did not increase their anxiety. Anxiety grew at a similar rate between accelerated and nonaccelerated honors students, but accelerated regular students increased their anxiety significantly faster than nonaccelerated regular students. Once students were accelerated, most variation in rates of attitude and anxiety change was at the student rather than school level. Racial ethnic background was the most important factor influencing rate of change at the student level. School contextual characteristics were major factors influencing rate of change at the school level. There is a lack of empirical knowledge concerning early acceleration of students in mathematics classes, particularly its impact on the development of motivational response, such as attitudes toward mathematics and mathematics anxiety. In the present study, I classified students into gifted, honors, and regular students to compare accelerated and nonaccelerated students in their attitudes toward mathematics and mathematics anxiety. In addition, the study identified student and school characteristics that promote or constrain the development of attitudes and anxiety among both accelerated and nonaccelerated students in each academic group. ACADEMIC ACCELERATION AS EDUCATIONAL PRACTICE Academic acceleration embodies the notion of instructional flexibility based on individual abilities without regard for age (Paulus, 1984, p. 98). In practice, academic acceleration takes the forms of early entrance to school, grade skipping, fastpaced classes in certain school subjects, advanced placement in certain school subjects, and college courses for high school students (Swiatek, 1993). Academic acceleration has been most often used with intellectually gifted students. In academically gifted students, academic acceleration has shown some positive effects (see Benbow, 1991; Southern & Jones, 1991; Stanley, 1991; Swiatek, 1993; Van TasselBaska, 1992). In particular, academic acceleration appears to help gifted students establish academic interests and build a strong foundation for future learning (Swiatek, 1993). One advocate (Kulik, 1992) argued that subjectspecific acceleration is one of the most effective ways to meet the academic needs of gifted students. Nevertheless, concerns about early acceleration of students exist, including gaps in knowledge and skills, poor retention of knowledge and skills, affective difficulties (e.g., anxiety, stress, and depression), and the possibility of being burned out both physically and mentally (Van TasselBaska, 1992). Cooperative learning and curriculum enrichment are alternative practices to acceleration that allow educators to tailor the curriculum to meet the differential needs of students (Karper & Melnick, 1993). Liu and Liu (1997) contended that students benefit more from curricular enrichment that emphasizes divergent thinking, originality, and problem solving than from acceleration. ATTITUDES TOWARD MATHEMATICS AND MATHEMATICS ANXIETY There is a history of work on attitudes toward mathematics, which Neale defined in 1969 as a multifaceted construct that describes a liking or disliking of mathematics, a tendency to engage in or avoid mathematical activities, a belief that one is good or bad at mathematics, and a belief that mathematics is useful or useless (p. 632). Similarly, the empirical record is extensive on mathematics anxiety, which Wood (1988) defined as the general lack of comfort that someone might experience when required to perform mathematically (p. 11). The values that students hold about any educational pursuit can be important to its success (Updegraff & Eccles, 1996). In forming these values, students draw heavily on their attitudes and anxieties (McLeod, 1992). According to McLeod (1994), students construct attitudes and anxieties about subject matters in the same ways that they form identities or come to understand other aspects of their world. This process is subject to many individual and environmental influences, such as student, family, and school characteristics. The impact of early acceleration of students in mathematics on the development of their attitudes toward mathematics and mathematics anxiety is largely unknown. One study by Durden and Tangherlini (1993) documented significant improvement in attitudes toward mathematics and motivation to learn mathematics among accelerated students. Based on the same survey data, Miller, Mills, and Tangherlini (1995) emphasized that accelerated students particularly improved their interest in and confidence with mathematics significantly. Another study by Kolitch and Brody (1992) reported that during the 1st year of college more than 90% of students who were accelerated in mathematics in secondary school indicated that they planned to major in mathematics or science. These accelerated students participated in college level mathematics and science courses at levels that were at least equal to those of peers who were not accelerated in mathematics in secondary school (Swiatek & Benbow, 1991a, 1991b). Thus, some research has found that accelerated students in mathematics not only maintain an interest in mathematics but also improve their education pursuits. The present investigation further investigated this hypothesis, using data from the Longitudinal Study of American Youth (LSAY). I assessed direct relationships between acceleration in secondarylevel mathematics and measures of attitudes toward mathematics and mathematics anxiety in students from the LSAY. METHOD The LSAY is a national 6year (19871992) panel study of mathematics and science education in public middle and high schools in the United States (Miller & Hoffer, 1994). The LSAY was designed to study students academic and nonacademic growth over time and the community, family, school, and classroom factors that promote or inhibit student growth. A national probability sample of 52 schools participated in the LSAY. Approximately 60 seventh graders selected from each of these schools across the United States were followed for 6 years (Grades 7 to 12). The sample consisted of 3,116 students. The present study used the entire sample of students. MEASURES AND VARIABLES Attitudes toward mathematics and mathematics anxiety were outcome variables constructed from selfreport measures included in the comprehensive educational survey of the LSAY (see Miller & Hoffer, 1994, for rationale and construction of these measures). Based on prior research, these scales were determined by LSAY researchers to be valid indicators of essential aspects of mathematics attitudes and anxiety. Attitudes scores were represented in a metric of 016 with a higher value indicating a higher level of attitudes for the present investigation. Cronbachs alphas across Grades 7 to 12, respectively, were 0.69, 0.66, 0.67, 0.72, 0.76, and 0.74. Anxiety scores were represented in a metric of 08 with a higher value indicating a higher level of anxiety for the present investigation. Cronbachs alphas across Grades 7 to 12, respectively, were 0.62, 0.68, 0.71, 0.71, 0.71, and 0.75. These reliability coefficients were deemed adequate (see Davies & Cummings, 1998). Composite variables of attitudes and anxiety were constructed for the present study from items on each scale, and as reported previously the measures produced adequate reliability coefficients. Background variables included gender (coded as 1 for female and 0 for male), age, number of parents (coded as 1 for single parent and 0 for two parents), number of siblings, mothers and fathers education (years of education), mothers and fathers socioeconomic status (SES; continuous variables constructed from occupations), and home language (coded as 1 for other languages than English and 0 for English). Because race/ethnicity had five categories, four binary variables were created to represent Hispanic, Black, Asian, and Native American with White as the baseline category (against which other categories were compared). Correlations of all background variables were small (shared variances were less than 25%), except for the correlation between Asian race/ethnicity and home language other than English (shared variance was 49%). Even this largest correlation was not exceedingly high, indicating no serious collinear problems among background variables. Background variables were either standardized to have a mean of 0 and a SD of 1 (for mothers and fathers SES) or centered around the grand mean (for all other variables). To obtain estimates for what is often referred to as the typical student with nationally average characteristics, all background variables need to be centered (see Bryk & Raudenbush, 1992). Centered variables reflect proportions (e.g., the typical student attends a school with a nationally average proportion of female students). Table 1 presents means and standard deviations for all major school variables represented in the study. Among school context variables, the percentage of students given free lunch was used to represent school socioeconomic composition, and the percentage of minority students represented school racial composition. With three categories (urban, suburban, and rural), two binary variables (suburban and rural) were created to represent school location, with urban as the baseline category (against which suburban and rural were compared). Some school climate variables were composites, showing adequate Cronbachs alphas at the school level (0.83 for academic expectation, 0.87 for disciplinary climate, 0.88 for principal leadership, 0.85 for teacher autonomy, 0.78 for general support for mathematics, and 0.70 for extracurricular activities). No serious collinear concerns arose among school variables (the largest correlation, 0.67, between academic expectation and principal leadership was not exceedingly high). As with student characteristics, school variables were centered for analysis. STATISTICAL PROCEDURES Many researchers prefer to use IQ scores to identify gifted students. However, there were no IQ measures obtained in the LSAY. Given a strong correlation in previous research between IQ and standardized achievement test scores, students were grouped according to their mathematics achievement in Grade 7 to form three academic categories for the research: gifted, honors, and regular students (see Lim, 1995). The 90th percentile was used to identify gifted students. In line with Schack (1993), who speaks against treating the rest of the students as a whole group of regular students, the remaining students formed two groups. Honors students were those with mathematics achievement scores between the 65th and 90th percentiles; regular students fell below the 65th percentile. This classification is slightly different from that of Schack (1993), who placed students below the 55th percentile as special needs students. The threeway classification I used avoids defining special education based on academic achievement. Early access to advanced mathematics courses is a common form of early acceleration of students in mathematics. One approach is to provide students with early access to Algebra I at the beginning of junior high school (i.e., Grades 7 and 8). In the present study, for each of the three academic categories (gifted, honors, and regular), students who took Algebra I in either Grade 7 or Grade 8 were considered accelerated in mathematics, whereas those who took Algebra I in neither Grade 7 nor Grade 8 were considered not accelerated in mathematics. Therefore, the crossclassification between academic group (three categories) and acceleration status (two categories) created six groups of students (e.g., gifted students accelerated, gifted students not accelerated). The threelevel hierarchical linear model (HLM) served as the primary statistical technique (Bryk & Raudenbush, 1992). I developed 12 HLM models, one for each group of students to model attitudes toward mathematics and one to model mathematics anxiety. The following description of the HLM model used attitudes as an example. The same description applies to anxiety. The first level of the HLM is a withinstudent model with one set of linear regression equations (one for each student). Each regression equation modeled a students attitudes scores on his or her grade levels (Grade 7 was defined as time zero). As a result, each student had an estimated rate of change in attitudes from Grade 7 to Grade 12. The second level of the HLM is a betweenstudent model with one set of linear regression equations (one for each school). Each regression equation modeled students individual rates of change in attitudes (from the first level) using studentlevel variables. As a result, each school had an estimated average rate of change in student attitudes from Grade 7 to Grade 12 adjusted for student characteristics in that school. The third level of the HLM is a betweenschool model with one linear regression equation modeling schools average rates of change in attitudes (from the second level) using schoollevel variables. A simple HLM model was first estimated without betweenstudent and betweenschool variables. The rate of change in attitudes was described as an average value (fixed effect) plus a variation (random effect). In HLM this estimate is made at both student and school levels, thus offering an opportunity for researchers to divide variance for the rate of change in attitudes into student and school components with adjustments for sampling and measurement errors. In addition, the relationship between the rate of change in attitudes and the status in attitudes at time zeroGrade 7 (often referred to as initial status)could be examined. A complex HLM model was then developed with betweenstudent and betweenschool variables included. The purpose was to use these variables to explain variation between students (within schools) and between schools in the rate of change in attitudes, and thus the complex model determined salient student and schoollevel variables that affected the rate of change in student attitudes across the three academic categories. RESULTS Following the presentation of descriptive statistics, results are presented in the previously described order of these HLM analyses. STUDENT ATTITUDES TOWARD MATHEMATICS AND MATHEMATICS ANXIETY Table 2 presents attitudes and anxiety scores across grade levels. Among 276 gifted students, 141 (from 32 schools) were not accelerated in mathematics, and 135 (from 26 schools) were accelerated (about 49%). Although both accelerated and nonaccelerated gifted students showed a decline in attitudes and a growth in anxiety across Grades 7 to 12, differences between them were no greater than 0.5 points in either attitudes or anxiety, a trivial difference given that the attitudes scale is 016 and the anxiety scale is 08. Thus, both accelerated and nonaccelerated gifted students showed similar attitudes and anxiety levels at each grade. Among 701 honors students, 556 (from 49 schools) were not accelerated in mathematics, and 145 (from 31 schools) were accelerated (about 21%). Although accelerated honors students showed slightly higher attitudes and slightly lower anxiety scores than honors students who were not accelerated at each grade level (with the exception of anxiety scores in Grade 8), their differences were no larger than 0.5 points in either attitudes or anxiety (with the exception of attitudes scores in Grade 7). Therefore, accelerated honors students showed similar attitudes and anxiety levels to those shown by honors students who were not accelerated at each grade. Among 1,813 regular students, 1,744 (from 50 schools) were not accelerated in mathematics, and 69 (from 24 schools) were accelerated (about 4%). Gaps in attitudes and anxiety scores between accelerated and nonaccelerated regular students were relatively larger compared with those of gifted and honors students. Accelerated regular students showed consistently higher attitudes and consistently lower anxiety scores than regular students who were not accelerated across grade levels, with differences ranging from 0.59 to 1.32 points in attitudes and from 0.30 to 0.80 points in anxiety. Therefore, results were marginally in favor of accelerated regular students at each grade level. Descriptive results based on raw data without any statistical adjustment over sampling and measurement errors provided some preliminary ideas about the development of attitudes and anxiety among students in each academic group. However, sampling and measurement errors were taken into account in the results from the HLM models, which provided more reliable estimates of the development of attitudes and anxiety. VARIANCE IN THE RATE OF CHANGE IN ATTITUDES TOWARD MATHEMATICS AND MATHEMATICS ANXIETY Results from the simple HLM models presented in Table 3 show the extent to which students and schools were responsible for the variation in the rate of change in attitudes and anxiety. There was a statistically significant variation in the rate of change in attitudes among accelerated gifted students. Although there was a statistically significant variation in the average rate of change among schools attended by these accelerated gifted students, schools accounted for only about 11% of the variation. For gifted students who were not accelerated about 47% of the variation was attributable to schools. The schools of accelerated honors students contributed to the variation in their rates of attitudes change (variance attributable to schools was significant at about 39%). For honors students who were not accelerated about 15% of the variation was observed at the school level. For accelerated regular students the entire variation in the rate of change in attitudes was attributable to student effects (no statistically significant variation was attributable to differences between schools). For regular students who were not accelerated about 15% of the variation was observed at the school level. The distribution of variance in the rate of change in anxiety is also reported in Table 3. For accelerated gifted students about 26% of the variation was observed at the school level. For gifted students who were not accelerated about 72% of the variation was observed at the school level. For both honors and regular students who were accelerated the entire variation in the rate of change in anxiety was attributable to studentlevel effects. For honors and regular students who were not accelerated about 77% and 24%, respectively, of the variation was observed at the school level. RATE OF CHANGE IN ATTITUDES TOWARD MATHEMATICS AND MATHEMATICS ANXIETY Estimates both on the initial (Grade 7) status and on the rate of change in attitudes and anxiety are presented in Table 4. The differences between accelerated students and those who were not accelerated in each academic category were tested for statistical significance using 95% confidence intervals (Glass & Hopkins, 1984). At the beginning of junior high school (Grade 7), both accelerated and nonaccelerated gifted students showed statistically significant positive attitudes toward mathematics. Initial status in attitudes, however, was not significantly different between those accelerated (12.71) and those who were not accelerated (12.81). Both accelerated and nonaccelerated gifted students experienced a statistically significant decline in attitudes from Grade 7 to Grade 12. On average the rate of decline was about 0.30 points each grade level among accelerated gifted students and about 0.21 points each grade level among gifted students who were not accelerated. This difference in the rate of decline in positive attitudes was not statistically significant. Correlation between initial status and rate of change is an important indicator of the nature of the change. A positive correlation shows a fanopen pattern, whereas a negative correlation shows a fanclose pattern (Bryk & Raudenbush, 1992). In the case of attitudes, a fanopen (fanclose) pattern indicates that the variation among students in attitudes increased (decreased) from Grades 7 to 12, with students maintaining their relative positions and students with a higher initial status in attitudes showing a slower (faster) rate of decline in positive attitudes from Grades 7 to 12. Correlation between average initial status and average rate of change can be interpreted in the same manner at the school level. There was a fairly weak fanclose pattern among gifted students (both accelerated and nonaccelerated), but a strong fanclose pattern at the school level for both accelerated and nonaccelerated gifted students. Both accelerated and nonaccelerated honors students showed statistically significant positive initial status in attitudes. However, the rate of decline in attitudes was statistically significant for both accelerated and nonaccelerated honors students. On average the rate of decline was about 0.34 points each grade level among accelerated honors students and about 0.31 points each grade level among honors students who were not accelerated. This difference was not statistically significant. Initial status was not related to rate of decline among either accelerated or nonaccelerated honors students, but there was a strong fanclose pattern at the school level for both accelerated and nonaccelerated honors students. Both accelerated and nonaccelerated regular students showed statistically significant positive initial status in attitudes that favored accelerated students. The rate of decline in attitudes was statistically significant among both accelerated and nonaccelerated regular students. On average the rate of decline was about 0.43 points each grade level among accelerated regular students and about 0.29 points each grade level among regular students who were not accelerated. Accelerated regular students showed a significantly faster rate of decline in positive attitudes than regular students who were not accelerated. A strong fanopen pattern was observed at the student and school levels for accelerated regular students, but there was a fairly weak fanclose pattern at the student and school levels for regular students who were not accelerated. Both accelerated and nonaccelerated gifted students shared similar initial (Grade 7) status in anxiety, and there was no statistically significant change in anxiety for either accelerated or nonaccelerated gifted students across Grades 7 to 12. There was a fairly weak fanclose pattern among accelerated and nonaccelerated gifted students. But there was a strong fanclose pattern at the school level for both accelerated and nonaccelerated gifted students. Both accelerated and nonaccelerated honors students had similar initial status in anxiety, and both showed a statistically significant growth in anxiety across Grades 7 to 12. On average the rate of growth was about 0.12 points each grade level among accelerated honors students and about 0.08 points each grade level among honors students who were not accelerated. The rate of growth in anxiety was not significantly different for the two groups. There was a strong fanopen pattern among accelerated honors students but a trivial fanclose pattern among those who were not accelerated. However, there was a strong fanclose pattern at the school level for both accelerated and nonaccelerated honors students. Accelerated regular students had significantly lower anxiety than regular students who were not accelerated starting junior high school, but showed a rate of growth in anxiety about 0.16 points each grade level, significantly higher than the rate of growth about 0.06 points each grade level among regular students who were not accelerated. There was a strong fanopen pattern at the student and school levels for accelerated regular students but a fairly weak fanclose pattern at the student and school levels for regular students who were not accelerated. VARIABLES RELATED TO RATE OF CHANGE IN ATTITUDES TOWARD MATHEMATICS Table 5 shows the HLM results for student and schoollevel variables related to rates of change in attitudes and in mathematics anxiety. To illustrate the magnitude of each effect and compare effects across different groups of students, a common metric was used. I report statistical results in effect size (SD) units. Rosenthal and Rosnow (1984) classified effect sizes of more than 0.50 SD as large, between 0.30 and 0.50 SD as moderate, and less than 0.30 SD as small. The magnitude of the effects in Table 5 was explained according to these standards. For accelerated gifted students, White students showed a significantly faster rate of decline in attitudes than did Native American students (1.28 SD). This large effect indicated that gifted Native American students (compared with gifted White students) were fairly resistant to the decline in attitudes. Older students showed a significantly faster rate of decline in attitudes than did younger students (0.07 SD). Cumulatively, to arrive at a moderate effect size, an age difference of 4 years is needed (this is rare at the same grade level). This age effect was trivial. At the school level gifted students in schools with a high percentage of freelunch students declined in attitudes at a significantly faster rate than those in schools with a low percentage of freelunch students (0.59 SD). This is a large effect of school socioeconomic composition. For gifted students who were not accelerated, White students showed a significantly faster rate of decline in attitudes than did Black students (3.46 SD). At the school level gifted students in schools where mathematics teachers assigned more mathematics homework declined in attitudes at a significantly faster rate than those in schools where mathematics teachers assigned less mathematics homework (0.72 SD). For accelerated honors students, Asian students declined in attitudes at a significantly faster rate than White students (1.17 SD). Students with fewer siblings at home showed a significantly faster rate of decline in attitudes than those with more siblings at home (0.42 SD). At the school level students in schools with a narrow grade span declined in attitudes at a significantly faster rate than those in schools with a broad grade span (0.31 SD). Students who attended schools with weak principal leadership showed a significantly faster rate of decline in attitudes than those who attended schools with strong principal leadership (1.20 SD). For honors students who were not accelerated, there were statistically significant gender differences in the rate of decline in attitudes (0.33 SD in favor of males). Older students declined significantly faster in attitudes than younger students (0.03 SD). Students with both parents showed a significantly faster rate of decline in attitudes than those with single parents (0.54 SD). At the school level students attending schools with fewer extracurricular activities showed a significantly faster rate of decline in attitudes than those attending schools with more extracurricular activities (0.29 SD). For accelerated regular students, no studentlevel variables were found to be responsible for the decline in attitudes. Because there was no statistically significant variation in the average rate of decline in attitudes among schools attended by these accelerated regular students, no schoollevel variables were used in the model. For regular students who were not accelerated, female students declined in attitudes at a significantly faster rate than male students (0.21 SD), and White students showed a significantly faster rate of decline in attitudes than Black students (0.75 SD). Students whose home language was English showed a significantly faster rate of decline in attitudes than those whose home language was other than English (0.29 SD). At the school level students attending small schools showed a significantly faster rate of decline in attitudes than those attending large schools (0.13 SD). Students who attended schools with strong principal leadership showed a significantly faster rate of decline in attitudes than those who attended schools with weak principal leadership (0.45 SD). VARIABLES RELATED TO RATE OF CHANGE IN MATHEMATICS ANXIETY For accelerated gifted students, older students showed a significantly faster rate of growth in anxiety than younger students (0.03 SD). At the school level students in schools with a low teacher education level grew in anxiety at a significantly faster rate than those in schools with a high teacher education level (0.98 SD). For gifted students who were not accelerated, White students showed a significantly faster rate of growth in anxiety than Black students (3.24 SD). At the school level students in schools where fewer students shared a computer for mathematics education grew in anxiety at a significantly faster rate than those in schools where more students shared a computer for mathematics education (2.19 SD). For accelerated honors students, both Black and Asian students showed a significantly faster rate of growth in anxiety than White students (3.15 SD and 2.45 SD, respectively). Students whose fathers had low SES showed a significantly faster rate of growth in anxiety than those whose fathers had high SES (0.70 SD). Because there was no statistically significant variation in the average rate of growth in anxiety among schools attended by these accelerated honors students, no schoollevel variables were used in the model. For honors students who were not accelerated, older students showed a significantly faster rate of growth in anxiety than younger students (0.13 SD). At the school level students in schools where fewer students shared a computer for mathematics education grew in anxiety at a significantly faster rate than those in schools where more students shared a computer for mathematics education (1.40 SD). Students in schools with fewer extracurricular activities grew in anxiety at a significantly faster rate than those in schools with more extracurricular activities (0.29 SD). For accelerated regular students, students whose home language was English showed a significantly faster rate of growth in anxiety than those whose home language was other than English (3.69 SD). Because there was no statistically significant variation in the average rate of growth in anxiety among schools attended by these accelerated regular students, no schoollevel variables were used in the model. For regular students who were not accelerated, no studentlevel variables were found to be responsible for the growth in anxiety. At the school level students in rural schools grew in anxiety at a significantly faster rate than those in urban schools (0.98 SD). DISCUSSION In the present study, students in each of three academic categories (gifted, honors, and regular) were either accelerated (took Algebra I in either Grade 7 or Grade 8) or not accelerated (took Algebra I in neither Grade 7 nor Grade 8). When gifted and honors students were accelerated, the development of their attitudes toward mathematics was found to be similar to that of their counterparts in the same academic category. These accelerated gifted and honors students were not disadvantaged in the development of their attitudes compared with their counterparts who were not accelerated. However, when regular students were accelerated in mathematics, the development of their attitudes was significantly disadvantaged compared with regular students who were not accelerated (attitudes of accelerated regular students declined significantly faster than that of regular students who were not accelerated). When gifted students were accelerated, their mathematics anxiety did not increase over time (neither did the anxiety of gifted students who were not accelerated). Gifted students in early acceleration in mathematics did not become more anxious about mathematics. When honors students were accelerated, their anxiety increased significantly over time but in a similar amount in comparison with honors students who were not accelerated. Honors students in early acceleration did not increase anxiety beyond that typical among honors students who were not accelerated. When regular students were accelerated, their anxiety increased at a significantly faster rate over time in comparison with regular students who were not accelerated. Regular students accelerated early became substantially more anxious about mathematics than regular students who were not accelerated. Therefore, for gifted and honors students, early acceleration in mathematics does not necessarily confer negative motivational or emotional effects. On the other hand, for regular students, early acceleration in mathematics may have negative effects. Attitudes of accelerated regular students may decline and mathematicsrelated anxiety of these students may grow at a significantly faster rate than those of regular students who are not accelerated. Regular students who were not accelerated mirror honors students who were not accelerated in terms of decline in attitudes and growth in anxiety. This finding suggests that if regular students are not accelerated, the development of their attitudes and anxiety can be similar to that of students in the honors category. Overall, findings of the present study call for caution when planning to accelerate regular students in mathematics. These results for accelerated regular students can be interpreted using the theory of affectivecognitive consistency. The theory holds that there is consistency between affective and cognitive components of attitudes, and when inconsistency occurs, there is a tendency for one or the other to change (Triandis, 1971). The cognitive challenge is obvious for accelerated regular students who engaged in a faster pace or at an advanced level of learning with some handicaps in mathematics abilities (as reflected in their Grade 7 mathematics achievement). According to Festinger (1957), this cognitive inconsistency is not comfortable, and students will take steps to increase their comfort level and redress the inconsistency. A decline in attitudes toward mathematics and a growth in mathematics anxiety are thus natural (and effective) affective reactions to the cognitive inconsistency caused by early acceleration in mathematics. Regular students who are not accelerated do not experience as great a cognitive imbalance as accelerated regular students. Therefore, their decline in attitudes and growth in anxiety would not be expected to be as great as those of accelerated regular students. The theory of affectivecognitive consistency also applies to the situation for accelerated gifted and honors students. With superior abilities in mathematics, these students are much better prepared cognitively for early acceleration in mathematics than are accelerated regular students. Although these students also react to reduce cognitive inconsistency, their decline in attitudes and growth in anxiety are not exceedingly greater than those among gifted and honors students who were not accelerated. Once students (in particular regular students) are accelerated in mathematics, counseling services should be made available to them. If the person receives a communication that changes his [or her] cognition, he [or she] may restructure these cognitions so that new distinctions are made among them, and his [or her] affect may not change (Triandis, 1971, pp. 7273). It appears that a condition for early acceleration of students in mathematics is that students are interested in and confident with mathematics. Once students experience the faster pace or the advanced level of learning in acceleration, they need to be directed to make a clear distinction between some costs of mathematics and overall benefits of mathematics. This distinction makes students continue to feel positive about mathematics overall but disappointed in some aspects of mathematics in their particular situation. Because there is little cognitive change with this distinction, there is little need for affective change (Triandis, 1971). A SPECIAL LOOK AT ACCELERATED GIFTED STUDENTS One study examined attitudes and attitude changes among mathematically talented students selected for an acceleration programthe University of Minnesota Talented Youth Mathematics Program (UMTYMP). As in the present study, Terwilliger and Titus (1995) observed significant declines in attitudes on most scales over the first 2 years among accelerated talented students. But the degree of the decline in attitudes shown in the present study is quite different from the sharp decline in attitudes reported in Terwilliger and Titus (1995, p. 29). In particular, they noted a substantial decline in confidence in mathematics among accelerated talented students. If confidence in mathematics mirrors mathematics anxiety, then the present study has shown a quite different phenomenon: There was no growth in anxiety across junior and senior high school among accelerated gifted students. Some studies have reported favorable affective outcomes for gifted students in early acceleration. Despite their decline in attitudes, students selected for the UMTYMP [still] have generally more positive attitudes than do those who were not selected (Terwilliger & Titus, 1995, p. 31). Compared with their peers, students selected for the Model Mathematics Program (MMP) significantly improved their attitudes toward mathematics (interest and confidence) and motivation to learn mathematics (Durden & Tangherlini, 1993; Miller et al., 1995). The present study, on the other hand, failed to identify any motivational or affective advantages for accelerating gifted students. Findings of the present study are closer to those of other studies of early acceleration of gifted students. Students in the Study of Mathematically Precocious Youth (SMPY) reported similar satisfaction with schooling and themselves to that of abilitymatched students who were not accelerated (Swiatek & Benbow, 1991a, 1991b). Students in the Gifted Mathematics Program (GMP) showed similar preferences for activities in the mathematics classroom to gifted students who were not accelerated, such as working on interesting problems, participating in competition, and learning about ideas outside textbooks (Robinson & Stanley, 1989). Despite some specific discrepancies (probably due to different samples and measures), all these pieces of evidence support one common conclusion about early acceleration of gifted students in mathematics. They do not appear to burn out because of early acceleration; from a motivational and affective perspective, they do quite well. In fact, accelerated gifted students fare as well as, if not better than, gifted students who are not accelerated on the motivational and affective outcomes observed. Once again, the theory of affectivecognitive consistency may explain this phenomenon. VARIANCE DISTRIBUTION IN ATTITUDES TOWARD MATHEMATICS AND MATHEMATICS ANXIETY Once students (gifted, honors, and regular) were accelerated in mathematics, most variation in their rate of decline in attitudes toward mathematics and their rate of growth in mathematics anxiety was found at the student rather than school level. Student rather than school factors were thus responsible for the variation in the rate of change in attitudes and anxiety. This common pattern across two affective outcomes and three academic categories may also be explained using the theory of affectivecognitive consistency. Achieving a balance (consistency) between affective and cognitive components is something done by individual students. Schools, of course, can use communication to intervene, but communication for such a purpose has not been a part of any traditional early acceleration programs in mathematics. On the other hand, school effects on the rate of change in attitudes and anxiety were stronger for gifted, honors, and regular students who were not accelerated (i.e., there were larger proportions of variance attributable to schools). This finding, together with the previous one, suggests that when achieving a balance between affective and cognitive components is overwhelmingly the work of students themselves, they appear to become insensitive to external influences. In contrast, when achieving a balance is not left only to students, schools are in a position to influence the rate of change in attitudes and anxiety. GRADE 7 STATUS AND RATE OF CHANGE IN ATTITUDES TOWARD MATHEMATICS AND MATHEMATICS ANXIETY A striking finding of the present study is the strong fanopen pattern among accelerated regular students in attitudes toward mathematics and mathematics anxiety at the student and school levels. Accelerated regular Grade 7 students who had less favorable attitudes and higher anxiety showed a rapid decline in attitudes and a rapid growth in anxiety when compared with accelerated regular students of that age who had more positive attitudes and lower anxiety. Note that the strong fanopen pattern at the school level indicates that schools also added negative influence in this situation. These results contrast with those for regular students who were not accelerated in mathematics. Here results showed no meaningful relationship between Grade 7 status and the rate of change in attitudes and anxiety at either the student or the school level. One interpretation of these findings is that I have identified the most vulnerable group of students with this analysisthat is, those who are at risk of affective distress when provided with early acceleration in mathematics. A different story emerges for accelerated gifted and honors students: A strong fanclose pattern across attitudes and anxiety was found at the school level. These data suggest that if educators wish to ensure that accelerated gifted and honors students fare well affectively, their attention might profitably be directed at schools rather than individuals. Students in schools with favorable average affective outcomes in Grade 7 were actually at high risk of affective distress under conditions of early acceleration. Again, the theory of affectivecognitive consistency may apply here. Overall, the strong pattern (either fanopen or fanclose) indicates that the rate of change in attitudes and anxiety can be reasonably predicted from Grade 7 status in attitudes and anxiety. Strong patterns such as those reported in this paper have often been used to predict later learning problems from initial learning status (Willms & Jacobsen, 1990). SIGNIFICANT STUDENT CHARACTERISTICS FOR SUCCESSFUL EARLY ACCELERATION The examination of studentlevel effects creates a good opportunity to identify students who are winners from the perspective of attitudes toward mathematics and mathematics anxiety once they are accelerated in mathematics. If I assume that student characteristics with large effects (effect size greater than 0.50 SD) are deemed key individual characteristics of the winners, I find that when gifted students were accelerated, Native American students fared well affectively (their rate of decline in attitudes was the lowest). When honors students were accelerated, White students (particularly White students whose fathers had high SES) had positive affective profiles (their rate of decline in attitudes and their rate of growth in anxiety were the lowest). When regular students were accelerated, students whose home language was other than English fared well affectively (their rate of growth in anxiety was the lowest). A comparison of these winning characteristics of students indicates the need to better examine racial/ethnic differences in the development of attitudes and anxiety. If home language is a reasonable reflection of raceethnicity, then the winning student characteristics are all racial/ethnic in nature. Presumably, these racial/ethnic differences are related to cultural influences. Students affective reactions to mathematics occur within a larger framework of how students make sense of their world in general (McLeod, 1994, p. 644), and this framework is highly socialcultural (Bishop, 1991, 2000). In relation to the theory of affectivecognitive consistency, the present study suggests that cultural values may play an important role in the rebalancing of affective and cognitive components. Terwilliger and Titus (1995) reported that among those who were selected for the [UMTYMP] program, boys showed a significantly higher level of motivation, confidence, and interest in mathematics than did girls (p. 29). But the present study found no significant gender differences in either Grade 7 status in attitudes or in the rate of decline in attitudes among accelerated gifted students. Although accelerated male (or female) gifted students may show some substantial local fluctuation in attitudes (2 years in their study), gender differences appear to be absent in the development of attitudes across junior and senior high school. SIGNIFICANT SCHOOL CHARACTERISTICS FOR SUCCESSFUL EARLY ACCELERATION The examination of schoollevel effects helps identify effective schools where students fare well from the perspective of attitudes toward mathematics and mathematics anxiety once they are accelerated in mathematics. Again, I can assume that school variables with large effects are key characteristics of effective schools. When gifted students were accelerated, those in schools with a favorable socioeconomic composition and favorable teacher preparation (education levels) fared well affectively (school average rate of decline in attitudes and school average rate of growth in anxiety were the lowest). When honors students were accelerated, those in schools with strong principal leadership were outstanding from the affective perspective (school average rate of decline in attitudes was the lowest). School effects were smaller on the rate of change in attitudes and anxiety when regular students were accelerated. Two out of three school characteristics are contextual. School context appears to play an important role from the affective perspective in early acceleration of students in mathematics. This finding is in line with the conceptual framework of school contextual effects (see Sammons, 1999; Teddlie & Reynolds, 2000). Being deeply entrenched through residential segregation and various forms of selective schooling, school context often plays a particularly strong role in many educational outcomes, such as mathematics achievement (Rumberger & Willms, 1992). Educators who intend to practice early acceleration in mathematics may wish to examine their school context critically (e.g., school resources and teacher qualification). Studies examining early acceleration of students in mathematics rarely consider how schools influence educational outcomes of accelerated students. The focus of most studies is individual psychosocial adjustment of students given early acceleration (see Swiatek & Benbow, 1991b). The present study makes a contribution by showing that school characteristics can have effects on the rate of change in attitudes and anxiety over and above the effects of student characteristics (i.e., there were school effects over and above individual differences in the rate of change in attitudes and anxiety). In fact, the number of large school effects is the same as the number of large student effects in the data, which indicates that to some extent school effects may be just as important as individual differences. RECOMMENDATION FOR FURTHER RESEARCH The present study shows linear changes across junior and senior high school in attitudes towards mathematics and mathematics anxiety. Some research has shown that measures of anxiety are related curvalinearly to educational outcomes, particularly when anxiety is considered a component of motivation (Lens & Decruyenaere, 1991). This challenge to the linear notion of change in anxiety should be considered when designing further studies. A nonlinear specification can help pinpoint the turning point from which abnormal (accelerated) decline in attitudes and growth in anxiety begin among accelerated gifted, honors, and regular students, and such results will inform intervention and modification of acceleration programs. This research was supported by a grant from the American Educational Research Association, which receives funds for its AERA Grants Program from the National Science Foundation and the U.S. Department of Educations National Center for Education Statistics and the Office of Educational Research and Improvement under NSF Grant #RED9452861. Opinions reflect those of the author and do not necessarily reflect those of the granting agencies. REFERENCES Benbow, C. P. (1991). Meeting the needs of gifted students through use of acceleration. In M. C. Wang, M. C. Reynolds, & H. J. Walberg (Eds.), Handbook of special education: Research and practice (Vol. 4, pp. 2336). Elmsford, NY: Pergamon. Bishop, A. J. (1991). Mathematical enculturation: A cultural perspective on mathematics education. Boston: Kluwer. Bishop, A. J. (2000). Critical challenges in researching cultural issues in mathematics learning In M. L. Fernandez (Ed.), Proceedings of the 22nd Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 2329). Columbus, OH: ERIC. Bryk, A. S., & Raudenbush, S. W. (1992). Hierarchical linear models. Newbury Park, CA: Sage. Davies, P. T., & Cummings, E. M. (1998). Exploring childrens emotional security as a mediator of the link between marital relations and child adjustment. Child Development, 69, 124139. Durden, W. G., & Tangherlini, A. E. (1993). Smart kids: How academic talents are developed and nurtured in America. Seattle, WA: Hogrefe & Huber. Festinger, L. (1957). A theory of cognitive dissonance. Stanford, CA: Stanford University Press. Glass, G. V., & Hopkins, K. D. (1984). Statistical methods in education and psychology. Englewood Cliffs, NJ: PrenticeHall. Karper, J., & Melnick, S. A. (1993). The effectiveness of team accelerated instruction on high achievers in mathematics. Journal of Instructional Psychology, 20, 4954. Kolitch, E. R., & Brody, L. E. (1992). Mathematics acceleration of highly talented students: An evaluation. Gifted Child Quarterly, 36, 7886. Kulik, J. A. (1992). An analysis of the research on ability grouping: Historical and contemporary perspectives. Storrs, CT: National Research Center on the Gifted and Talented. Lens, W., & Decruyenaere, M. (1991). Motivation and demotivation in secondary education: Student characteristics. Learning and Instruction, 1, 145159. Lim, T. K. (1995). Using the SATM to identify the mathematically talented in Singapore. Gifted Education International, 11, 3438. Liu, Z., & Liu, P. (1997). Practice and theoretical exploration on supernormal education of mathematics to gifted and talented children from 68. In J. Chan, R. Li, & J. Spinks (Eds.), Maximizing potential: Lengthening and strengthening our stride. Proceedings of the 11th World Conference on Gifted and Talented Children (pp. 130135). Hong Kong: University of Hong Kong. McLeod, D. B. (1992). Research on affect in mathematics education: A reconceptualization. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 575596). New York: Macmillan. McLeod, D. B. (1994). Research on affect and mathematics learning in the JRME: 1970 to the present. Journal for Research in Mathematics Education, 25, 637647. Miller, J., & Hoffer, T. B. (1994). Longitudinal Study of American Youth: Overview of study design and data resources. Dekalb: Social Science Research Institute, Northern Illinois University. Miller, R., Mills, C., & Tangherlini, A. (1995). The Appalachia model mathematics program for gifted student. Roeper Review, 18, 138141. Neale, D. C. (1969). The role of attitudes in learning mathematics. Arithmetic Teacher, 16, 631640. Paulus, P. (1984). Acceleration: More than grade skipping. Roeper Review, 7, 98100. Robinson, A., & Stanley, T. D. (1989). Teaching to talent: Evaluating an enriched and accelerated mathematics program. Journal for the Education of the Gifted, 12, 253267. Rosenthal, R., & Rosnow, R. L. (1984). Essentials of behavioral research: Methods and data analysis. New York: McGrawHill. Rumberger, R. G., & Willms, J. D. (1992). The impact of racial and ethnic segregation on the achievement gap in California high schools. Educational Evaluation and Policy Analysis, 14, 377396. Sammons, P. (1999). School effectiveness: Coming of age in the twentyfirst century. Lisse, the Netherlands: Swets & Zeitlinger. Schack, G. D. (1993). Effects of a creative problemsolving curriculum on students of varying ability levels. Gifted Child Quarterly, 37, 3238. Southen, W. T., & Jones, E. D. (1991). Academic acceleration: Background and issues. In W. T. Southern & E. D. Jones (Eds.), The academic acceleration of gifted children (pp. 128). New York: Teacher College Press. Stanley, J. C. (1991). An academic model for educating the mathematically talented. Gifted Child Quarterly, 35, 3642. Swiatek, M. A. (1993). A decade of longitudinal research on academic acceleration through the Study of Mathematically Precocious Youth. Roeper Review, 15, 120124. Swiatek, M. A., & Benbow, C. P. (1991a). A tenyear longitudinal followup of participants in a fastpaced mathematics course. Journal for research in mathematics Education, 22, 138150. Swiatek, M. A., & Benbow, C. P. (1991b). Tenyear longitudinal followup of ability matched accelerated and unaccelerated gifted students. Journal of Educational Psychology, 86, 528538. Teddlie, C., & Reynolds, D. (2000). The international handbook of school effectiveness research. London: Falmer. Terwilliger, J. S., & Titus, J. C. (1995). Gender differences in attitudes and attitude changes among mathematically talented youth. Gifted Child Quarterly, 39, 2935. Triandis, H. C. (1971). Attitude and attitude change. New York: Wiley. Updegraff, K. A., & Eccles, J. S. (1996). Course enrollment as selfregulatory behavior: who takes optional high school math courses? Learning & Individual Differences, 8, 239260. VanTasselBaska, J. (1992). Educational decision making on acceleration and grouping. Gifted Child Quarterly, 36, 6872. Willms, J. D., Jacobsen, S. (1990). Growth in Mathematics skills during the intermediate years: Gender differences in school effects. International Journal of Educational Research, 14, 157174. Wood, E. F. (1988). Math anxiety and elementary teachers: What does research tell us? For the Learning of Mathematics, 8(1), 813. XIN MA is an associate professor and Director of the Canadian Centre for Advanced Studies of National Databases at the University of Alberta. He is Fellow of the National Academy of Education. His main research interests include mathematics education, school effectiveness, policy analysis, and quantitative methods. He is the author of the book A National Assessment of Mathematics Participation in the United States: A Survival Analysis Model for Describing Students Academic Careers (1997, Edwin Mellen).





