发布时间：2021-07-07 10:43

Keywords Mathematics identity Self-other discrepancy Self-versus reﬂected appraisals STEM coursetaking Mathematics achievement Symbolic interaction The authors thank Peter Burke and Richard Serpe for their helpful comments on an earlier d

Keywords Mathematics identity · Self-other discrepancy · Self-versus reﬂected appraisals · STEM coursetaking · Mathematics achievement · Symbolic interaction

The authors thank Peter Burke and Richard Serpe for their helpful comments on an earlier draft of the paper. * George W. Bohrnstedt GBohrnstedt@air.org 1 American Institutes for Research, Washington, USA

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1 Introduction

With job growth projected to continue across the world in science, technology, engi- neering and mathematics (STEM), preparing students for these jobs is a priority for many nations. In the United States, for example, a recent report documents a pro- jected overall 6.5% growth for jobs in science, technology, engineering, and math- ematics (STEM) in the ten years between 2014 and 2024. Growth is projected in all areas of STEM save for the sector that includes drafters, engineering technicians, and mapping technicians, where a small decline is projected (Fayer and Lacey,

2017).

Virtually all STEM jobs, whether in statistics, engineering, physics or the life sci- ences, require a ﬁrm knowledge of mathematics. While a solid education in math- ematics should begin in elementary school, it is high school mathematics achieve- ment that determines whether a student is prepared for a STEM major in college. For this reason, it is important to understand what leads to solid mathematics achieve- ment by the end of high school. Increasing the understanding of what leads to math- ematics achievement by the end of high school is the overall focus of this study, and in particular, the role that mathematics identity and the discrepancy between self- and others’ perceptions of being a “mathematics person” play in that achievement.

When considering the analysis of mathematics achievement by the end of high school, mathematics coursetaking obviously plays a central role (e.g., Byun et al., 2015). It is not only the courses taken that are important but the diﬃculty of these courses as well. For example, Lee et al. (1997) identiﬁed eight mathemat- ics course-taking patterns and showed that the level of diﬃculty of the courses taken was strongly correlated with academic achievement in mathematics. Simi- larly, Ma and Wilkins (2007) found that taking upper-level mathematics courses (trigonometry, precalculus, and calculus) was associated with student growth in mathematics tests performance.

But recent studies have also shown the signiﬁcant role that social psycho- logical factors play in academic achievement, and especially the role that aca- demic identities play (Marsh, 1990, 1993; Marsh et al., 1988). In a recent study, Bohrnstedt et al. 2020 showed the importance of mathematics identity for high school mathematics achievement. This study builds oﬀ the Bohrnstedt et al. 2020 study underscoring the important role that mathematics identity plays in grade 12 mathematics achievement. Both studies use the High School Longitudinal Study (HSLS:09), a study using a random sample of U.S. high schools and 9 th grad- ers in them as well as the same measure of mathematics identity; both also take into account previous mathematics achievement. However, the Bohrnstedt et al. 2020 study did not have students’ transcript data available and instead relied on students’ reports of mathematics courses taken in high school. In contrast, the current study uses data from students’ high school transcripts to determine cour- setaking in mathematics and science. In addition, the transcript data provides the students’ grade-point averages (GPAs) in the STEM courses taken—data not available when the Bohrnstedt et al. (2020) study was in process.

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Mathematics identity and discrepancies between self-and…

Finally, the current study also examines the role that a discrepancy between self and reﬂected appraisals with respect to being “a mathematics person” plays in high school mathematics achievement—a construct that was not examined in the earlier research. Speciﬁcally, is identity veriﬁcation (where self- and perceived appraisals by others are in agreement) more important than a lack of such veriﬁ- cation for mathematics achievement? Regarding the latter, does a discrepancy in self-appraisals versus reﬂected appraisals with respect to “being a mathematics person” aﬀect motivation in pursuit of mathematics achievement more than or less than the motivation provided by being veriﬁed by others? And if so, does it matter whether one appraises oneself more positively than others are perceived as appraising oneself compared to the perception that others appraise oneself more positively than one appraises oneself? Which is the more motivating?

In summary, the current study examines the roles that mathematics identity and discrepancies in self versus reﬂected appraisals as to whether one is “a mathematics person” play in mathematics achievement in high school. This leads to two research questions: 1. Does mathematics identity in grade 11 relate to mathematics achievement at grade 12? 2. Does a discrepancy between self-appraisals and reﬂected appraisals as to whether one is “a mathematics person” relate to grade 12 mathematics achieve- ment? And if it does, does the direction of the discrepancy matter in the relation- ship?

These two research questions are examined taking into account student mathe- matics and science coursetaking, STEM GPA, previous mathematics achievement as well as student background characteristics (gender, race-ethnicity, and SES).

2 Theoretical background

Research Question 1 Does mathematics identity in grade 11 relate to mathematics achievement at grade 12?

Identities can be classiﬁed into three types—social, role, and personal (Burke and Stets, 2009). Social identities are associated with membership categories such as gender, race, and class; role identities are associated with roles embedded in the larger social structure; and personal identities are associated with personal charac- teristics (e.g., smart, punctual, introverted). The focus in this study is on an iden- tity associated with the student role— mathematics identity. Does one see oneself as”a mathematics person” and do others also perceive one as being “a mathematics person?”.

From the symbolic interactionist perspective, identities grow out of the various roles played in the life cycle and are seen as growing out of interactions with sig- niﬁcant others. Based on one’s abilities and aspirations, one engages in role-related

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interactions that result in actual as well as perceived appraisals from signiﬁcant others (e.g. peers, parents, teachers, friends). These appraisals are important in the development of identities. In the symbolic interaction literature, it is the per- ceived appraisals rather than the actual appraisals that are important (Cooley, 1902; Mead, 1934).1 Our perceptions of signiﬁcant others’ attitudes toward the self, called “reﬂected appraisals,” are seen as providing evidence about how well one is per- forming in a given role. The better the performance and its associated reﬂected appraisals, the more valued the role is likely to become and the more likely it will become an identity. The import of this theoretical approach is that simply claiming an identity does not make it so. According to Stone (1962), identities are established when signiﬁcant others use the same words to describe someone as a person uses for him- or herself. Thus, to be identiﬁed, claims made for oneself must be legitimated and supported by signiﬁcant others. For example, a student’s mathematics identity is based not only on self-perceived mathematics abilities and accomplishments, but also on the reﬂected appraisals of others. In this regard, Wenger (1998) found that one’s perception of his or her mathematics identity was impacted by others’ percep- tions and evaluations in one’s immediate community of others, e.g. one’s signiﬁcant others. This in turn inﬂuenced one’s participation within that community. More gen- erally, the research evidence shows that how parents, teachers, and friends view one in relation to mathematics achievement has an impact on one’s perceptions of one’s mathematics competence (Robnett et al., 2018; Bleeker and Jacobs, 2004; Bouchery and Harter, 2005; Cribbs et al., 2015).

McCall and Simmons (1966, 1978) and Stryker (1968) argue that our various identities form an identity hierarchy which in turn deﬁnes the self. The aﬀective strength of our identities deﬁnes their prominence in the identity hierarchy (Brenner et al., 2017) and their salience the probability that the identity will be activated in a given role-related situation (Stryker, 1968). So, for example, if one is taking a rigor- ous mathematics course in high school and has a strong mathematics identity (it is prominent in the hierarchy) the probability is high that that identity will be activated and will result in highly motivated role-related behaviors (Burke and Stets, 2009; McCall and Simmons, 1978)—behaviors that maximize goal attainment, whether that goal be solving a particular mathematics problem, raising one’s hand to answer the teacher’s questions or achieving a high grade in the course.

Academic identities have long been considered important for understanding stu- dents’ persistence and academic success (Marsh, 1990, 1993; Marsh et al., 1988).

Identities are seen as having a strong motivational component—they propel one to either seek out (or to avoid) situations where the identity can be reinforced or strengthened (e.g., entering -or avoiding- a mathematics competition). In other cases, one can ﬁnd oneself in situations that automatically activate the identity which then plays a positive role in the quality of a role-related outcome (e.g., having

1 This does not mean that actual feedback is unimportant; indeed, it is incorporated into our perceptions.

But there is less than a perfect correlation between signiﬁcant others’ actual appraisals and our percep- tions of them. See Lundgren (2004) who ﬁnds a moderate relationship based on a review of the research literature.

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Mathematics identity and discrepancies between self-and…

to take a classroom mathematics test and doing well in it). More recently, research- ers have shown the importance of how content-speciﬁc identities (e.g., mathematics identity, science identity, engineering identity) not only motivate academic perfor- mance, but also how they relate to STEM-related occupational choice (Carlone and Johnson, 2007; Cass et al., 2011; Chemers et al., 2011; Godwin et al., 2013; Hazari et al., 2009; Stets et al., 2017; Syed et al., 2011). All these ﬁndings suggest that the stronger the mathematics identity, the greater the likelihood of aspiring to or choos- ing a STEM occupation.

Research Question 2 Does a discrepancy between self- and reﬂected appraisals of whether one is “a mathematics person” relate to grade 12 mathematics achieve- ment? And if it does, does the direction of the discrepancy matter in the relationship?

Charles H. Cooley (1902) developed the notion of the “looking glass self”— our “reﬂection” of how we think others see us and this perspective was further developed by Mead (1934). More formally, we see ourselves through the reﬂected appraisals of others. We seek to have our self-views conﬁrmed by others in our com- munities of involvement, whether home, school, the playground, voluntary associ- ations, informal get-togethers, or the workplace. Not having such conﬁrmation of one’s identity can lead to personal upset and actions to attempt to seek consonance between self- and other evaluations (Burke and Stets, 2009). Again, it is our per- ceived appraisals from others, not their actual appraisals that are important. The fact that they are perceived in no way invalidates their importance, since it is perceptions that provide agency to much of our behavior and account for feelings of accomplish- ment or upset, whether those perceptions are accurate or not. Perceptions whether accurate or not are real in their consequences.

There have been several research studies that have examined the eﬀects of a mis- match between one’s identity standard—where the identity standard contains the meanings that deﬁne how we see ourselves—our self-appraisals (Burke and Stets, 2009)—and our reﬂected appraisals.2 The working assumption is that we seek iden- tity veriﬁcation—that is, we seek congruence between our self-appraisals and those

2 There is also research literature in psychology based on Higgins’ (1987) self-discrepancy theory. Hig- gins (1987) examines discrepancies between actual self, ideal self, and ought self from the perspectives of both the self and others. Discrepancies in the actual-ideal self are hypothesized to lead to depressed aﬀect, while actual-ought discrepancies are hypothesized to lead to anxiety. While identity and self-dis- crepancy theories are similar in some ways, as Large and Marcussen (2000) point out, identity theory assumes that an individual can have an identity for each role occupied, self-discrepancy theory does not incorporate any speciﬁc identity, but rather deals with the self as a whole. In a theoretical treatise, Large and Marcussen (2000) suggest how identity and self-discrepancy theories can be integrated in a way that allows for a more precise prediction of the kind of emotional response (depression versus anxi- ety) that results from discrepancies that occur in social interactions by considering the roles that oughts and obligations play for each speciﬁc identity one holds. There has been little research building oﬀ the integrated theory of Large and Marcussen (but see Marcussen, 2006). However, research studies follow- ing from Higgins’ (1987) self-discrepancy approach include outcomes as disparate as depression (Scott and O’Hara, 1993), psychological disturbances (Heidrich and Powwattana, 2004), personality ratings (Funder, 1980), procrastination (Orellana-Damacela et al., 2000), moods (Tangney et al., 1998), and decision-making (Krishnamurthy and Kumar, 2002).

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of signiﬁcant others. Much of the research in this area has demonstrated the negative eﬀects that mismatches have on the self, on the one hand, and the positive feelings that go with identity veriﬁcation on the other. When mismatches occur, research shows that persons who are unable to achieve congruity between their self-stand- ards and their reﬂected appraisals experience stress (Zanna and Cooper, 1976). The assumption is that the lack of self-veriﬁcation leads to feelings of distress followed by attempts to re-achieve congruity. This rebalancing can take several forms. One of the ways mentioned by Burke and Stets (2009) is selective aﬃliation—that is, inter- acting with others who provide feedback that is consistent with our self-views. A second is altercasting—a strategy by which we place others in situations or identities such that we increase the likelihood of them providing support for our self-views (Weinstein and Deutschberger, 1963). For example, one might represent oneself as being “down” to a signiﬁcant other because of a bad role performance (e.g., one gets a bad mathematics test score) with the hope and expectation that the signiﬁcant other will characterize the performance as being out of character (“I know you’re really good at math, but you had an ‘oﬀ’ day.”). While identity veriﬁcation is moti- vating because of the positive feeling that results from it, non-identiﬁcation is also motivating because of the negative feelings that non-identiﬁcation carries with it.

Burke and Stets (1999), in a study of married couples, found that couples who mutu- ally veriﬁed each other’s spousal identities over time expressed increases in love, trust, and commitment to the marriage. On the other hand, when the spousal identity was not conﬁrmed, the level of stress associated with the relationship increased.

One might assume that the greater the discrepancy in a negative direction between self and reﬂected appraisals, the more positive the eﬀect. That is, one might expect that when others regard our role-related performances more positively than we view them ourselves, the result would be contentment and happiness. But the research literature suggests that the relationship is more complex than that. For example, Burke and Harrod (2005), in examining data from their longitudinal study of newly married couples, found, as might be expected, that a respondent was likely to be distressed if one’s self-view was not reciprocated by one’s spouse. But surpris- ingly, both negative and positive discrepancies resulted in increased stress. Indeed, they found that the greater the discrepancy in either direction, the higher the prob- ability of divorce. When our self-views are consistently not being reciprocated by signiﬁcant others, one strategy is to remove oneself from the oﬀending situation—in this case, through divorce.

Recent research by Stets and Burke (2014), Trettevik (2016), and Stets and Trette- vik (2016) has further reﬁned the relationship between positive and negative self- and reﬂected appraisals: there appears to be a positive response to small negative discrepancies between self- and reﬂected appraisals, but large discrepancies in either direction result in feelings of distress or, as in the case of a recent study by Stets and Trettevik (2016), unhappiness. That is, the relationship is non-linear and is captured best by an inverted U-shape. However, when the outcome is behavioral instead of aﬀective, the relationship between the behavioral outcome and discrepancies in self- and reﬂected appraisals is linear, not curvilinear (Burke & Stets, 2009).

Given the outcome in the current study is mathematics achievement—a behav- ioral rather than an aﬀective outcome—the assumption is that the relationship will

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Mathematics identity and discrepancies between self-and…

be a linear one. However, the direction of the discrepancy could be of importance as well. There are two logical, possible predictions that could be made. Stets et al. (2017) in a study that examined the role of having a science identity on entering a science occupation hypothesized that negative discrepancies (where reﬂected appraisals of a one as a science student are more positive than one sees oneself in that regard) would be negatively related to the pursuit of a science career. Their rationale was that when signiﬁcant others’ views are more positive than one’s own, it sets up expectations that are too high for one to meet and reduces motivation to pursue the career. In the case when one’s perception of oneself as a science student is more positive than the reﬂected appraisals of others, the prediction is that one will be more likely to pursue a science career in order “to counteract the discrep- ancy” (Burke and Stets, 2009: p. 9) between self and signiﬁcant others’ appraisals.

By achieving this outcome, the presumption is that signiﬁcant others’ views will rise to match one’s own and such congruence leads to self-veriﬁcation (Burke and Stets,

2009).

But the opposite prediction could also be made. That is, increased motivation could also follow from the situation where signiﬁcant others’ perceived apprais- als are more positive than one’s own appraisals; the hypothesis in this case is that believing that signiﬁcant others hold one in higher esteem than one holds oneself with regard to identity-related role-playing leads one to try harder as a way to meet the expectations of one’s signiﬁcant others, which in turn leads to better outcomes compared to the situation where self-appraisals are more positive than reﬂected appraisals. That is, a negative discrepancy could “add a boost” to the motivation to perform well on an identity-related task (e.g., taking a mathematics assessment); and doing well is hypothesized to lead to an increase in one’s self-appraisal thereby leading to self-veriﬁcation.

What does the research literature tell us about which of these two hypotheses appears to be the correct one? The Stets et al. (2017) study suggests that neither of these explanations may be correct since their study found there was no relationship between the self- and reﬂected appraisals’ discrepancy with respect to being a sci- ence person and choosing a science career. The current study is important to fur- ther examine whether a self-reﬂected appraisals discrepancy with respect to being “a mathematics person” matters when mathematics achievement is the outcome, or whether, as found in the Stets et al. (2017) study there is no relationship.

3 High school coursetaking and mathematics achievement

Several studies have examined the relationship between the diﬃculty of courses taken and future academic performance. Mathematics courses are more orderable by diﬃculty than any other subject, where each course typically requires comple- tion of courses at the immediately previous level. Lee et al. (1997) identiﬁed eight mathematics course-taking patterns and showed that the level of diﬃculty of the courses taken was strongly correlated with academic achievement in mathematics.

Studies have typically applied coding anchored at the lower end by general math- ematics and, at the high end, by advanced courses such as calculus (Adelman, 1999;

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Schneider et al., 1998; Schiller and Hunt, 2003). Ma and Wilkins (2007) also found that taking upper-level mathematics courses (trigonometry, precalculus, and calcu- lus) was associated with student growth in mathematics test performance. By con- trast, science course sequences appear to be cumulative rather than sequential. The more science courses a student takes regardless of the order in which they are taken, the better the student is positioned to enroll in college (Adelman, 2006).

Indeed, research studies have found that STEM course-taking in high school is strongly related to students’ performance on high-stakes achievement tests (Allen, 2015; Ma and Wilkins, 2007; Madigan, 1997; Noble and Schnelker, 2007; Noble et al., 2006). Noble and Schnelker (2007) examined the eﬀects of course-taking pat- terns on students’ ACT performance in English, mathematics, and science, taking prior performance into account. Taking advanced courses such as trigonometry, pre-calculus, and calculus—in addition to algebra I, geometry, and algebra II—was associated with the highest ACT mathematics scores. Allen (2015), in a technical brief on subject-speciﬁc ACT performance, found that the strongest predictor of per- formance on ACT science was end-of-course chemistry achievement.

Similar ﬁndings are also reported for low-stakes assessments—assessments where the results are not reported to the student or anyone else, nor are they used for grad- ing, graduation, or any other similar purpose. National assessments such as the U.S.

National Assessment of Educational Progress (NAEP) fall into this category, as does the Trends in International Mathematics and Science Study (TIMSS). Leow et al. (2004), analyzing TIMSS data using propensity score matching methods, found that consistent advanced-course-taking was associated with higher TIMSS achievement.

In another study, Byun et al. (2015) investigated the eﬀect of advanced mathematics course-taking—beyond Algebra II—on achievement and college enrollment using longitudinal and nationally representative data from Education Longitudinal Study of 2002 (ELS:2002). Both achievement and enrollment were shown to be positively associated with advanced course-taking.

Taken together, these results underscore the importance of course-taking for aca- demic achievement and the strong role that mathematics and science course-taking plays. Therefore, in this study’s investigations of the relationship between math- ematics identity, identity discrepancies, and mathematics achievement, the anal- yses also include the diﬃculty and content of students’ mathematics and science course-taking.

4 Methodology 4.1 Data

The data used in this study come from a sample of students who participated in two United States-based studies: the High School Longitudinal Study of 2009 (HSLS:09) and the 2013 grade 12 NAEP mathematics assessment. The HSLS:09 is a nationally representative, longitudinal study of approximately 24,000 9th grad- ers in 940 schools, including both public and private schools. HSLS:09 is designed to collect information on student’s trajectories from the beginning of high school

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Mathematics identity and discrepancies between self-and…

into postsecondary education and, eventually, the workforce, by following the cohort until age 30. HSLS:09 included an algebra assessment and survey component in the fall of 9th grade (2009) and again in the spring of most students’ 11th grade year (2012). A follow-up data collection occurred in spring 2013, the students’ expected graduation year, to collect data on students’ postsecondary pursuits. Students’ high school transcripts were collected in 2014, the year after graduation, from their high schools. In addition, two items measuring mathematics motivation—deﬁned by HSLS:09 as mathematics identity along with other measures of motivation were administered at grades 9 and 11.

In spring 2013, some students participating in the HSLS:09 were also sam- pled to take the NAEP grade 12 mathematics and reading assessments in the main NAEP assessment testing window. This NAEP-HSLS overlap sample enables the estimation of relationships between NAEP achievement and variables collected in HSLS:09, including the mathematics motivation and transcript data that are the focus of this study. The overlap sample consisted of roughly 4,200 students, about 3,500 of which were administered the mathematics assessment in approximately 320 schools. These data allow us to combine information from the NAEP, the only national test of mathematics achievement in the United States that ﬁrst selects a nationally representative sample of schools and then randomly selects students within schools, with the rich data collected by the HSLS:09 including the students’ transcripts. NAEP reports on 4th and 8th graders’ achievement in mathematics and reading every two years, as well as reporting periodically on 12th graders’ achieve- ment in mathematics and reading, as well as on other subject areas. NAEP math- ematics measures ﬁve content areas that are combined into an overall score as well as being available as subscores: (a) number properties and operations; (b) measure- ment; (c) geometry; (d) data analysis, statistics, and probability; and (e) algebra.3 Combining the subscores into an overall score is justiﬁed by the high intercorrela- tions among the subscores; they range from 0.91 to 0.95.4 Of the roughly 3500 students who were included in both the HSLS:09 sample and the 2013 NAEP grade12 Mathematics sample, about 2600 had complete data on all the variables. In the full HSLS:09 sample, about 14,700 observations had similarly complete data. The descriptive socio-demographic statistics for the full HSLS:09 and the overlap sample are shown in the Appendix, Table 3. There were some small diﬀerences in the two samples on the race/ethnicity variable; students in the over- lap sample appear to have slightly higher socio-economic status than those in the full HSLS:09 sample. The two samples are very similar with respect to the percent- age of male and female students. The overlap sample also diﬀers slightly from the full HSLS:09 sample in terms of STEM course-taking and achievement; students in the overlap sample were more likely to have enrolled in higher-level mathematics courses which is consistent with the overlap sample being of higher SES than the

3 At grade 12, the measurement and geometry subscores are combined for reporting purposes since most of the measurement material in the grade 12 assessment is geometric in nature.

4 Correlations can be found at: https:// nces. ed. gov/ natio nsrep ortca rd/ tdw/ analy sis/ 2013/ scali ng_ deter minat ion_ corre latio ns_ math2 013co nditi onal. aspx.

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full sample. There were also some small diﬀerences in the distribution of science courses taken in the two samples.

4.2 Measurement

The various constructs being measured can be divided into ﬁve categories: Grade 12 mathematics achievement (the outcome variable), mathematics motivation, self- versus reﬂected appraisals discrepancy, STEM coursetaking and performance, and background characteristics, the latter of which are taken into account in the analyses.

Each of these is discussed immediately below.

4.2.1 Grade 12 mathematics achievement

Grade 12 mathematics achievement is measured by the overall NAEP grade 12 mathematics score5 and is designated in the statistical analyses as Grade 12 math- ematics achievement (Yi), where (Yi) is used to designate the NAEP score as the dependent variable for the ith student in the sample.6 The overall NAEP score, which is what is used in this study, has a range of 0–300. The mean score on the 2013 assessment was 155.8 with a standard deviation of 28.1.

4.2.2 Mathematics identity

Mathematics identity (IDENTITY) is measured by a two-item index collected in the grade 11 HSLS:09 follow-up, the year prior to the assessment of NAEP mathemat- ics achievement. The two items are: “You see yourself as a mathematics person” (self-appraisal) and “Others see you as a mathematics person” (reﬂected appraisal).

Each item had four response categories: “strongly agree,” “agree,” “disagree,” and “strongly disagree,” originally coded 1, 2, 3, and 4 respectively, but for this study were recoded to 3,2,1,0 respectively so that the higher the value, the greater the agreement that one sees oneself as “a mathematics person,” or that one sees others as perceiving oneself as “a mathematics person.” The rationale for using these two items as a measure of mathematics identity follows from the theoretical develop- ment above where identity is comprised of both self- and reﬂected- appraisals. The internal consistency reliability for the index as measured by alpha (Cronbach, 1951) was 0.88.

5 For more information about the grade 12 NAEP mathematics assessment see: https:// www. natio nsrep

ortca rd. gov/ readi ng_ math_ g12_ 2013/#/.

6 NAEP scale scores are reported as 20 “plausible values” (plausible scores) because, by the NAEP assessment design, students are administered only a small subset of the total pool of assessment items.

This design is used to reduce student burden which would be excessive given the length of the NAEP

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