Abstract
The Mathematical Resilience Scale measures students’ attitudes toward studying mathematics, using three correlated factors: Value, Struggle, and Growth. The Mathematical Resilience Scale was developed and validated using exploratory and confirmatory factor analyses across three samples. Results provide a new approach to gauge the likelihood of student participation and persistence in mathematics.
Many influences in the life of a student converge in the mathematics classroom, resulting in some students continuing their pursuit of mathematics despite experiencing setbacks while others do not. Resilience is a desirable attribute for all students; it is the quality of being able to respond positively in the face of difficulties. Both the media and scholarly researchers have recently increased their interest in the relationship of noncognitive traits such as grit and resilience to educational outcomes (Duckworth, Peterson, Matthews, & Kelly, 2007; Tough, 2012; Yeager & Dweck, 2012). Yet there does not seem to be a consensus on the definition of resilience or how to study it. Resilience has been defined as a set of attributes, a process, and an outcome (Ahern, Kiel, Sole, & Byers, 2006). More generally, for resilience to be studied there has to be a positive response following some type of adversity (Luthar, Cicchetti, & Becker, 2000).
To provide context for the development of the concept of mathematical resilience, consider the experience of a hypothetical student in mathematics class. The student functions in an environment in which there is an expectation of growth in her knowledge of mathematics through her interactions with the teacher, other students, the content, and the curriculum. Now consider that this student may be of lower to average ability level in a heterogeneous classroom. The student observes the teacher presenting new mathematics content, solving problems, leading discussions and answering questions quickly and efficiently. In fact, effective teachers who are well prepared with strong content knowledge complete these tasks smoothly and almost effortlessly, in the eyes of the student (Borko & Livingston, 1989). The student may observe—and also compare themselves with—higher ability students who complete work and respond to questions from the teacher with relative ease and comfort (Chiu et al., 2008). Students of lower or average ability levels do not benefit from comparisons with students of higher ability levels (Reuman, 1989) and may experience these comparisons as reflective of an elitism described by a student in a study by Nardi and Steward (2003) as, “they just seem so clever” (p. 359). These experiences contribute to the development of an attitude that only people with high ability levels can succeed at mathematics (Nardi & Steward, 2003).
Based on the observations of this hypothetical student, mathematics looks like it comes with little effort to some people. Yet anyone who has pursued mathematics knows it does not come without struggle. The process of struggling is at the heart of learning mathematics, solving problems through combining many areas of cognitive functioning with trial and error. A visit to almost any school provides many opportunities to view inspirational posters displaying the quotes from many famous scientists and scholars on this topic. For example, Einstein (2010) said “Do not worry about your difficulties in Mathematics. I assure you mine are still greater” (p. 80). In a speech, Frederick Douglass (1857) said “If there is no struggle, there is no progress.” Here lies the cognitive disconnect for students who are challenged by mathematics: They are told that struggle is expected, yet they do not see anyone struggling except perhaps others like themselves.
This research reports the results of the development and validation of the Mathematical Resilience Scale (MRS), an affective instrument that measures a set of student attitudes related by theory to mathematical resilience. The MRS supports research on the impact of affective traits such as motivation and attitude on persistence in mathematics; on the interaction with other attitudes, values, and beliefs related to learning mathematics; and on the development of interventions to improve mathematical resilience. The intended use of the instrument is by university-level academic advisors, counselors, and researchers to better understand and differentiate students whose attitudes may contribute to a lack of persistence in mathematics especially when faced with challenges and setbacks. The ultimate goal of this research is to improve understanding of attitudes related to persistence in mathematics and improve student prospects for persistence and success in mathematics at all levels. Increased knowledge of the constructs assessed in this instrument can improve interventions to increase mathematical resilience in students and raise levels of achievement and ability to function mathematically (Johnston-Wilder & Lee, 2010).
Background
Mathematics as Part of Science, Technology, Engineering, and Mathematics (STEM)
Persistence in mathematics coursework is a key factor in improving student access and success in STEM careers. A study by the National Center for Educational Statistics (Chen, 2013) indicated that 48% of university students who chose a STEM major between 2003 and 2004 had changed their major by 2009, and half of these selected a major not considered STEM. Echoing the authors, more research is needed to better understand why students leave STEM majors. Research in motivational theory can improve our understanding of the problem and improve persistence in STEM careers (Perez, Cromley, & Kaplan, 2014). In addition, to encourage more students of all ability levels to persist in mathematics, new approaches to mathematics education should be explored. Mathematics is a key part of the STEM curriculum, arguably the most important discipline of the four because mathematics is foundational to the development of the majority of scientific theory and properties (Shaughnessy, 2012). New theories and interventions need to be identified and tested to encourage higher rates of student persistence in mathematics, enrollment in advanced mathematics, engagement in the content of STEM fields, and choice of STEM careers. Research has shown that affective traits such as motivation, beliefs, and attitude are linked to increased likelihood of taking advanced mathematics courses (Ma, 2006) and are significant predictors of improved cognitive activity and achievement (Buff, Reusser, Rakoczy, & Pauli, 2011; Ethington & Wolfe, 1986; Ma & Kishor, 1997).
Methods of Studying Resilience
Research on mathematical resilience borrows from psychological resilience, representing a positive response to a significant threat or adversity (Luthar et al., 2000). Psychological resilience provides a theoretical parallel, a language, and a research base for studying attributes that may be predictive of persistence in mathematics despite adversity. Resilience is often studied in people who have been exposed to dangerous, potentially life threatening, or stressful environments. All individuals are considered at risk when exposed to these threats; some respond with positive adaptations such as academic achievement, high school graduation, and college attendance while others suffer maladaptive symptoms such as involvement in drugs, crime, and academic failure. Luthar (2006) argued that because resilience is complex, it needs to be examined in the context of a particular domain, encouraging researchers to study many types of resilience including resilience in educational settings. Although psychological resilience has been researched extensively (Luthar, 2006; Luthar et al., 2000; Olsson, Bond, Burns, Vell-Brodrick, & Sawyer, 2003) the study of resilience in academic settings and specifically resilience in the study of mathematics represent a relatively new approach (Johnston-Wilder & Lee, 2010; Rivera & Waxman, 2011; Yeager & Dweck, 2012).
The construct of psychological resilience relates to an individual’s response following a significant adverse event. Resilience researchers focus on the outcomes (Olsson et al., 2003), the dynamic process (Luthar et al., 2000; Masten, 2001), or the set of protective factors that help an individual to cope (Karairmak, 2010). Resilient outcomes reflect successful life patterns such as emotional well-being, staying in school, and avoiding criminal activities (Olsson et al., 2003). Resilience as a process involves the analysis of interaction between the risk to the individual and the protective factors leading to an adaptive outcome which is identified as indicative of resilience. Protective factors reduce the effects of negative stimuli by helping an individual to persist through the challenges they face (Luthar et al., 2000). Protective factors include adaptive attitudes, supportive families, schools, communities, and peer groups that shield those at risk from negative outcomes. Studies focused on individual attributes predictive of resilience identify and measure personal characteristics that are consistently present in adaptive populations and absent in maladaptive populations (Ahern et al., 2006). The Connor–Davidson Resilience Scale (Karairmak, 2010) is an example of an instrument designed to measure affective traits that function as protective factors. Karairmak (2010) identified three factors on the Connor–Davidson Resilience Scale related to psychological resilience, two of which were considered relevant to mathematical resilience: tenacity and personal competence (e.g., “one works to attain one’s goals” and “I like challenge”) and tolerance of negative affect (e.g., one “tends to bounce back after illness or hardship” and one “knows where to get help”). These constructs are pertinent not only because of Karairmak’s work but also because of their relevance to the constructs in this study, discussed next.
Mathematical Resilience
Researchers are beginning to focus more attention on the response of students to difficult or challenging experiences while studying mathematics. Yeager and Dweck (2012) have studied resilience in a variety of contexts, defining resilience as any “behavioral, attributional, or emotional response to an academic or social challenge that is positive or beneficial for development” (p. 303). Using this definition, resilience can only exist in the aftermath of some type of adversity. In the context of mathematics, the adversity can take many forms, including a failing grade, struggle beyond the student’s tolerance, boredom, embarrassment due to poor performance, poor quality curriculum or instruction, and nonsupportive teacher–student or student–student interactions. The student may perceive that they are unable to learn the mathematics beyond some degree of rote memorization; they may have performance anxiety during class or on a test, or math anxiety (Chamberlin, 2010; Hembree, 1990; Richardson & Suinn, 1972; Tobias, 1978). That is, a student’s perception of the adversity is paramount to the study of resilience; the teacher’s or researcher’s perception of the experience is not relevant (Yeager & Dweck, 2012). The positive, resilient outcome can be proximal as in improved participation in the classroom, greater understanding of the material, enhanced interest and enjoyment, and improved performance on assessments, or distal as in persistence in taking higher levels of mathematics coursework and majoring in STEM careers.
Despite the complexity involved in studying resilience and the lack of a psychometrically tested instrument, there appears to be growing awareness of the importance of resilience for a student to be successful in mathematics (Johnston-Wilder & Lee, 2010; Rivera & Waxman, 2011; Yeager & Dweck, 2012). Johnston-Wilder and Lee (2010) suggested that mathematical resilience include perseverance despite setbacks, working collaboratively, the ability to articulate mathematical understanding, and a growth theory of learning. This study consisted of action research in an urban school in England in which the researchers were invited into the school to administer interventions for the purpose of improving student motivation and achievement levels in mathematics. Although Johnston-Wilder and Lee (2010) advocated the importance of mathematical resilience and provided an introductory theory, they used an instrument that had not been psychometrically evaluated. This limitation further supports a need for a mathematical resilience instrument. Rivera and Waxman (2011) conducted semistructured interviews of 118 English Language Learners to describe student perceptions and attitudes toward mathematics. Students in the study were considered at risk of academic failure by virtue of their status as English Language Learners. From this pool of students, teachers were asked to nominate students as resilient based on high achievement test scores, strong attendance, high-quality daily school work, and high level of motivation. Similarly, teachers were also asked to nominate students as nonresilient based on low test scores, attendance, school work, and motivation. The authors found significant differences between resilient and nonresilient students in attitudes and perceptions of mathematics. For example, students who were identified as resilient were more likely to say they liked math, were good at math, and would be able to receive help from parents on math. Although findings in Rivera and Waxman (2011) provide preliminary evidence of differences in attitudes between students identified as resilient and those who are not resilient, the subjective nature of the selection criteria diminished the strength of the results. In fact, the authors identified the use of teacher nomination of resilient students as a limitation of their study, further motivating development of an instrument.
Hypothesized Factor Structure and Theory
This study involves an investigation of the attitudes, beliefs, and values that distinguish a student who persists in mathematics from one that does not. The literature review develops the framework of mathematical resilience using current literature and provides insight into malleable attributes and protective factors that contribute to resilience.
Value
Value refers to the extent to which students find studying mathematics important in attaining their current or future goals, and is an established precursor to success (Deci, Vallerand, Pelletier, & Ryan, 1991). In addition, the extent to which a student perceives math to be valuable is believed to positively correlate with their level of motivation to study it (Eccles, 1983). Value is defined in terms of utility in a future career to align with the focus of the instrument for college students. College students are making choices that have a proximal impact on selection of coursework as well as a distal impact on major and future career.
Self-regulated learning also informs the Value construct. Self-regulation refers to the degree to which students can direct their efforts, skills, and abilities to achieve their desired academic goals (Zimmerman, 2008). Self-determination theory suggests that people self-regulate when they believe that their actions help them meet their needs (Ryan & Deci, 2000). University students have the freedom to choose their major, and for students considering STEM careers, this choice may lead to requirements to take extensive mathematics coursework. If a student greatly values either studying mathematics or careers that require mathematics, they are more persistent even in the face of challenge.
Struggle
Struggle refers to a student’s belief that they sometimes have to exert a great deal of effort because mathematics can be a challenge to learn, but their difficulty is not interpreted as an indication of personal incompetence. In other words, the experience of struggling with mathematics is not unusual; even exceptional mathematicians have to work hard and even make errors when learning and solving mathematics problems. The instrument focuses on Struggle in reference to student perceptions and tolerance of the level of difficulty in studying mathematics. When a student believes that Struggle is inherent to the study of mathematics, she attributes the reason for the challenge to aspects related to the mathematics rather than on limitations in her ability.
Several researchers have found differences between genders in students’ ability to deal with Struggle in mathematics. Licht and Dweck (1984) found that when higher ability level fifth-grade female students faced difficulty in mathematics, they performed worse than their male counterparts. Dweck (2007) also suggested that female students do not cope with the stress from the Struggle as well as male students, who were energized by it.
Recent research has shown more attention to the teachers’ and students’ attitudes toward struggling with mathematics. Autin and Croizet (2012) have also studied the effect of attitudes toward Struggle in learning and seem to have developed a promising intervention. They conducted three studies in which they reframed difficulty as an aspect of learning rather than a sign of intellectual incompetence. Results indicated that reframing improved cognitive performance by improving work memory capacity, providing evidence of the malleability of the construct defined as Struggle. L. M. Clark et al. (2014) studied mathematics teachers’ attitudes to investigate teacher sensitivity toward student dispositions and the connection with student learning. The factor named Teacher Allowance for Student Struggle with Problems measured teacher’s belief that learning mathematics should involve struggle (L. M. Clark et al., 2014, p. 258). Teachers with strong professional content knowledge also scored highest on the Teacher Allowance for Student Struggle with problems. L. M. Clark’s findings suggest that teachers who are more knowledgeable in mathematics have a greater appreciation of the struggle necessary to learn mathematics and can draw on this to improve student chances for success in mathematics.
The construct of Struggle is built on the theory of human agency, which purports that individuals assess and control their own thoughts, motivations, and actions (Bandura, 1989). One popular quote by Bandura provides the connection: “In order to succeed, people need a sense of self-efficacy, to struggle together with resilience to meet the inevitable obstacles and inequities of life” (Pajares, (n.d.) para. 6). Human agency is often exercised through the collective experiences and culture of a peer group. Bandura (2000) found that when the motivational investment of the group is high, the staying power in the face of setbacks is strengthened, hence enhancing performance. According to this theory, students who understand that Struggle in math is common to their peer group, to all mathematics students, and even to experts in mathematics have more tolerance and stronger staying power in the face of setbacks.
Growth
A growth theory of learning mathematics refers to the belief that people can improve their knowledge of mathematics. Implicit theories of intelligence, as reported in Dweck (1986, 2000) and Yeager and Dweck (2012), provide a model of either an incremental (growth) theory of intelligence or an entity (fixed) theory of intelligence. Students with a growth theory of intelligence believe that if they work at it, they can learn more. Students with a fixed theory believe that their level of intellect is static, and they are limited in developing this fixed ability level.
Recent research on the incremental theory of intelligence emphasizes the need for a certain mind-set to help overcome the challenges that are common in the life of a student (Dweck, 2006; Yeager & Dweck, 2012). In studies of resilience in academic and social settings, an incremental theory of intelligence paired with the mastery goal orientation (in which people are focused on developing their own expertise instead of on the way in which they are perceived) was associated with significant differences in students’ ability to respond favorably to challenges (Yeager & Dweck, 2012). Research has also shown that an individual’s theory of intelligence is malleable, that it predicts academic performance over time, and that a growth theory of learning is associated with improved achievement (Blackwell, Trzesniewski, & Dweck, 2007; Yeager & Dweck, 2012).
Resilience
This study was initialized with a fourth hypothesized attitudinal factor, Resilience. Contrasting with math anxiety, which has been studied extensively (Hembree, 1990; Richardson & Suinn, 1972; Tobias, 1978), mathematical resilience considers the attitudes inherent to those who experience stress in their studies but respond by functioning optimally instead of developing anxiety. Resilience in this study focused on resilience as an attitude, based on the psychological literature, defined as the positive response to a significant adversity. The measure of the Resilience factor focuses on a self-assessment of how one responds to adversity in the mathematics classroom. As described in the literature review, there are other ways of measuring Resilience. Examples include using outcomes such as the number of mathematics courses taken or for university students, whether they selected a mathematics-related major. Research that measures both attitudes and outcomes may be optimal because outcomes can be used to validate the relevance of the attitudes.
Validation Study
Exploratory and confirmatory factor analyses were used to better understand how the response patterns for individual items relate to a smaller number of common factors. Exploratory factor analysis identifies groups of items for which response patterns were similar to create a more parsimonious set of factors. The exploratory factor analysis produces evidence to support a factor structure indicating the items that load onto particular factors, to identify items with low-level loadings, and to identify multidimensional items. Following the exploratory factor analysis, a confirmatory factor analysis on a new sample tests the strength of the a priori factor structure restricted to unidimensional loading of items to factors (McCoach, Madura, & Gable, 2013).
Participants
The instrument was pilot tested in three stages using three different samples. First, the instrument was administered to a convenience sample of 262 participants (160 actuaries, 59 community college students). For the first study, the participants responded to 33 items related to the scales and 5 demographic questions. This sample was used to complete an exploratory factor analysis (EFA1). The results of EFA1 served to identify the factor structure, reduce the number of items on each scale, and identify other items that needed fine tuning. A revised instrument was administered to 603 participants composed entirely of students attending a large research-intensive university located in the northeast. The sample was randomly divided into two approximately equal sets. One sample was used to perform an external replication of the exploratory factor analysis, EFA2, as recommended by Osborne and Fitzpatrick (2012) to determine if the different sample and item revision affected the factor structure. The other sample was used to complete a confirmatory factor analysis (CFA) to test the fit of the data to the theoretical structure using Amos 18.0.0 (Arbuckle, 2009; Thompson, 2010).
All three samples contained demographic questions such as profession/major, gender, race, age group, and highest level of mathematics taken so far in their schooling from pre-algebra to more advanced than calculus. Respondents in all three samples were asked to assess their knowledge of mathematics using a Likert-type scale from 1 (low) to 10 (high), referred to as self-reported knowledge of mathematics (SRKM). The three samples were found to be similar with respect to the distribution by gender, χ2(2, N = 823) = 1.90, p = .38. As expected, the samples were dissimilar with respect to race, χ2(1, N = 823) = 10.69, p = .02, and highest level of mathematics, χ2(1, N = 823) = 350.24, p < .01. The samples from EFA2 and the CFA were not significantly different based on race, χ2(10, N = 570) = 4.62, p = .91, or highest level of mathematics, χ2(10, N = 570) = 3.08, p = .98 (see Table 1).
Demographic Distribution of Each Sample.
Note. Major is not available for EFA1. EFA1 = exploratory factor analysis 1; SRKM = self-reported knowledge of mathematics; STEM = science, technology, engineering, and mathematics.
Study 1: Content Validation and EFA1
Item Generation and Content Validation
A literature review guided the development of 46 items using a 7-point Likert-type scale of 1 (completely disagree) to 7 (completely agree). Seven categories were selected based on research suggesting that five to seven response options can be treated as continuous (Rhemtulla, Brousseau-Lieard, & Savalei, 2012). Items underwent content validation to determine their degree of fit to the constructs of interest, using guidelines from McKenzie, Wood, Kotecki, Clark, and Brey (1999). A jury of 11 subject matter experts met the criteria of knowledge of mathematics, education, or the actuarial profession, of which 2 did not reply. The remaining nine subject matter experts were asked to function as content validation jurors. These nine jurors comprised five professors, of which two were professors of mathematics education, one of mathematics, one of gifted education, and one of assessment, one PhD in gifted education, two measurement and assessment doctoral students in educational psychology, and one actuary.
The content validity of the instrument was evaluated using a questionnaire that provided jurors with information on mathematical resilience to assist them in assessing item quality. The questionnaire contained a description of mathematical resilience and definitions of the four hypothesized factors of Value, Struggle, Growth, and Resilience. The 46 items were masked in terms of hypothesized factor structure and then scrambled in order of presentation. Jurors were asked to indicate the best category fit for each item, including the option of “no category,” and to indicate the certainty of the placement, from 1 (not very sure) to 3 (very sure). Jurors were also asked to indicate how relevant they felt each item was for the category on a 3-point scale composed of L (low/no relevance), M (mostly relevant), and H (highly relevant). Finally, jurors were asked to provide qualitative feedback on things such as the definition of the construct or wording of items. Responses were compiled to identify items that should be excluded or reworded for the final survey instrument.
We initially eliminated items that received less than 80% consensus regarding placement in the proper factor, with a goal of obtaining a minimum of 8 items, and a maximum of 10 items for each factor. For Resilience, Struggle, and Growth, this process eliminated all but eight items, but for Value, no items were eliminated. Additional criteria were applied in a stepwise fashion, using juror quantitative and qualitative feedback, to identify items that were confusing, multidimensional, or unclear. The final set of items included 8 items in Struggle, Growth, and Resilience, and 9 Value items, for a total of 33 items. Table 2 presents a list of the initial 33 pilot items, as well as the results from EFA1.
Standardized Factor Pattern Coefficients, Structure Coefficients, and Communalities for the Four-Factor Principal Axis Factor Analysis of the Mathematical Resilience Scale, EFA1.
Note. Pattern coefficients given in boldface have values of .40 or greater and signify items loading primarily with that factor. Pattern coefficients less than .10 were suppressed. The communality of measured variables is given by h2.
Item was reworded in EFA2.
Exploratory Factor Analysis 1
A convenience sample of 262 was collected consisting mostly of insurance actuaries (63.2%) from across the United States and community college students (23.3%) living in the northeast, with 253 participants remaining after listwise deleting subjects for missing values. Table 1 presents the demographic breakdown for the samples from all three studies. Insurance actuaries are particularly of interest in research on those who persist in the study of mathematics despite experiencing difficulties. Insurance actuaries must pass as many as 10 exams that assess complex mathematical and statistical content to advance in their career. The pass rate on these exams is typically below 50%, and it is common for actuaries to take the exams multiple times before succeeding (Casualty Actuarial Society, 2011). To provide variation, the sample included community college student based on a hypothesis that these students are more likely to have had difficulties with mathematics and less likely to have developed attributes related to mathematical resilience. Undergraduate students and adults in a variety of professions comprised the remainder of the sample. Although the sample of actuaries provided important feedback on the scale, full validation of the scale requires a sample that is (a) heterogeneous with respect to the construct in question, (b) homogeneous with respect to other characteristics, and (c) demographically similar to the target population (Lackey & Wingate, 1998). For this reason, additional samples were gathered for Studies 2 and 3.
Descriptive statistics were run to determine whether sufficient variation exists for a factor analysis. Negatively worded items (Items G3 to G7) were reverse scored so that higher scores reflected a favorable stance. Mean item scores ranged from 4.38 to 6.50, and standard deviations ranged from 0.91 to 1.78, indicating good variation in ratings across the 7-point scale. We also collected data from enough participants to constitute a “fair” sample size according to the guidelines provided by Comrey and Lee (as cited in Pett, Lackey, & Sullivan, 2003, p. 48), resulting in a sample size-to-item ratio of 8 respondents per item.
An exploratory factor analysis was completed using oblique rotation to identify the structure of the instrument. An oblique rotation (direct oblimen, δ = 0) was selected because the factors were believed to be correlated. The initial step involved completing a preliminary analysis to determine the number of factors to extract. Eight different criteria were examined when considering the number of factors: the Kaiser–Guttman rule criteria of eigenvalues greater than 1, scree plot, minimum average partial (MAP) squared and fourth power, parallel analyses principal components analysis (PCA) means and percentiles, and parallel analyses principal axis factoring (PAF) means and percentiles (O’Connor, 2000). Following recommendations outlined in McCoach et al. (2013) and Pett et al. (2003), all eight criteria were examined diagnostically, but priority was assigned to results from the parallel analyses PCA and PAF along with the MAP analyses, each of which indicated four factors. After deciding on the number of factors, PAF was completed retaining items with pattern coefficients of .40 or greater on one factor and less than .30 on a second factor. In addition, any item with pattern coefficients greater than .30 on more than one factor was considered multidimensional. These items were omitted or revised in the subsequent instrument. Referring to the guidelines in L. A. Clark and Watson (1995), scale average and individual interitem correlations were compared with the range of .15 to .50.
Based on the evidence of a four-factor structure from PAF, PCA, and MAP, a four-factor model was selected. The criteria suggested by Pett et al. (2003) were used to establish acceptable measures of sampling adequacy: a statistically significant result for Bartlett’s test of sphericity, χ2(528, N = 253) = 3,948.41, p < .001, a Kaiser–Mayer–Okin (KMO) statistic of .88, a nonzero determinant of the correlation matrix, and the diagonals of the anti-image correlation matrix were all greater than .70. The resultant pattern matrix, structure matrix, and extraction communalities from PAF with oblique rotation are presented in Table 2. Eigenvalues for both EFA1 and EFA2 are provided in Table 3. Corresponding changes have been done in text also. Please verify if correct and appropriate.] The results were encouraging in that the patterns generally followed theory with a four-factor model explaining 44.5% of the variance in the items. Correlations between the factors were lower than expected, and it was surprising that correlations with the Resilience factor were negative. The correlations were as follows: Value and Struggle, .28; Value and Growth, .32; Value and Resilience, −.01; Struggle and Growth, .20; Struggle and Resilience, −.14; Growth and Resilience, −.22.
Eigenvalues for EFA1 and EFA2.
Note. EFA = exploratory factor analysis.
Internal Consistency Reliability Analyses for EFA1 and Resulting Decisions
Value scale
Nine items were hypothesized to be associated with Value, or the extent to which students find studying mathematics valuable for current and future goals. The factor analysis extracted 12 items on a single factor including all 9 Value items, 2 Resilience items, and 1 Growth item. The pattern coefficients for most items included in the Value scale reflect the hypothesized factor structure. In addition, pattern coefficients for four items hypothesized as part of other factors loaded on the Value factor, namely Items G8, R1, R3, and R8. The Item R8 had pattern coefficients above .40 on more than one factor, so it was dropped. The resulting scale contained 11 items including Items G8, R1, and R3, and all hypothesized Value items except Item V6, which had a pattern coefficient below the threshold. These items exhibited high levels of internal consistency (coefficient α = .91, 95% confidence interval [CI] = [.89, .93]). Individual items tended to be moderately associated (interitem correlation M = .48, minimum = .14, maximum = .79, SD = .15). Items V1, V2, V4, and V5 all referred to the importance of mathematics in achieving future goals, resulting in high interitem correlations.
Struggle scale
Containing eight items, the Struggle scale was defined as student perception and tolerance of difficulty in studying mathematics. All eight items and two additional Resilience items, Items R2 and R7, loaded onto this factor. It could be argued that Items R2 and R7 align with Struggle because they address how a student responds to experiences of discouragement and confusion. The scale containing the 10 items was reliable (coefficient α = .82, 95% CI = [.78, .85]). Although the average interitem correlation was within the desired range, the minimum was lower than desired (M = .32, minimum = .08, maximum = .53, SD = .10).
Growth scale
Eight items were developed to measure Growth for EFA1. The Growth scale refers to the belief that knowledge of mathematics is malleable and that mathematics ability can be improved with effort. Five items had high pattern coefficients on Growth with only low pattern coefficients on another factor. The pattern coefficients for the remaining three items indicated some multidimensionality. As a result wording of these items was revised in Study 2. The final scale contained seven items (including Items G1 and G6 but not Item G8) and exhibited acceptable levels of internal consistency reliability (coefficient α = .83, 95% CI = [.80, .86]) and moderate interitem correlations (M = .42, minimum = .26, maximum = .58, SD = .10).
Resilience scale
The Resilience factor was defined as a self-assessment of one’s response to a significant adversity when learning mathematics. Based on the criteria mentioned above, three of the eight items appeared to be multidimensional. In addition, four items that were hypothesized as associated with resilience had pattern coefficients greater than .40 on other factors. In retrospect, the problems with this factor may be due to the definition used to develop items for Resilience. The items asked the participants to assess themselves on whether they respond positively when facing a setback, wording that was similar to the Struggle items. Based on these results, the Resilience factor was removed. The resulting instrument focused directly on the attitudes and their relation to outcomes rather than on a self-assessment of how well one responds to setbacks.
Summary of Study 1
The content validation and exploratory factor analysis provide initial evidence for the validity of the MRS as a measure of Value, Struggle, and Growth with respect to the study of mathematics. The resultant model contained three factors with 27 items, 11 measuring Value, 10 measuring Growth, and 6 measuring Struggle. Based on the results, the Resilience factor and all items associated with it were removed. For comparative purposes, the extraction criteria (e.g., MAP, PAF) were reanalyzed using only Value, Struggle, and Growth items and were found to support the three-factor structure. A second factor analysis completed using PAF indicated that the items loaded as hypothesized except two—Item G1 appeared multidimensional and Item G8 loaded on Value instead of Growth. This three-factor solution explained 44.2% of the variance with high internal consistency (Growth coefficient α = .82, 95% CI = [.79, .85], Value coefficient α = .89, 95% CI = [.87, .91], and Struggle coefficient α = .78, 95% CI = [.73, .82]. Correlations between the factors were higher once the resilience items were removed. The correlations were as follows: Value and Struggle, .25; Value and Growth, .29; Struggle and Growth, .17. The resultant pattern matrix, structure matrix, and extraction communalities from PAF with oblique rotation are presented in Table 4.
Standardized Factor Pattern Coefficients, Structure Coefficients, and Communalities for the Three-Factor Principal Axis Factor Analysis of the Mathematics Resilience Scale, EFA1.
Note. EFA = exploratory factor analysis. Pattern coefficients given in boldface have values of .40 or greater and signify items loading primarily with that factor. Pattern coefficients less than .10 were suppressed. The communality of measured variables is given by h2.
Item was reworded in EFA2.
Study 2: Exploratory Factor Analysis 2 (EFA2)
Early on in this study, it was recognized that full validation of the scale would require a replication of the analysis with a sample that was heterogeneous with respect to the construct in question, homogeneous with respect to other characteristics, and demographically similar to the target population (Lackey & Wingate, 1998; Osborne & Fitzpatrick, 2012). The instrument was revised to improve wording and content coverage, using the results of EFA1. Items G1, G6, S3, S7, and S8 were revised and Item G8 was dropped. Three items were added to provide broader content coverage. Most notably, the Resilience items were omitted altogether due to reconsideration of the self-assessment factor after failing to develop a set of items that were unidimensional.
The resultant 27-item survey was administered to a second convenience sample of 603 collected from undergraduates attending a research-intensive state university located in the northeast. These students were enrolled in large lecture hall courses, including Philosophy, Art History, and Introductory Statistics. The sample was divided into two approximately equal groups using SPSS (SPSS Inc., 2009) to select random samples (n = 293 for EFA2 and n = 310 for the CFA), with 280 and 290, respectively remaining after list-wise deletion. A demographic breakdown of all participants is shown in Table 1. Undergraduate students were selected over younger students because of the availability of outcome variables that can be used to establish construct validity. For example, undergraduate students have already selected coursework that may include advanced mathematics, and many of them have already selected a major. These variables provide an indication that the student has persisted in studying mathematics, a desirable outcome for resilience research.
Descriptive statistics were run after reverse-scoring Items G3 to G7 and Item G9 to ease interpretation. The mean item scores ranged from 3.83 to 6.09 (SD between 1.10 and 1.75), indicating a reasonable range and good variation.
Following the same procedure as the EFA1, the same extraction techniques were applied using an oblique rotation to identify the structure of the instrument. The parallel analysis PCA and PAF and the MAP techniques all provided evidence of a three-factor structure. The sample was suitable for exploratory factor analysis based on Bartlett’s test of sphericity, χ2(351, N = 280) = 3,360.94, p < .001, the KMO statistic (.88), diagonals greater than .70 on the anti-image correlation matrix, and a nonzero determinant of the correlation matrix (Pett et al., 2003).
Three factors were extracted using PAF; the pattern matrix, structure matrix, and extraction communalities, generated for these items using extraction and oblique rotation, are presented in Table 5. As before, we retained items with pattern coefficients of .40 or greater on one factor and less than .30 on other factors (McCoach et al., 2013; Pett et al., 2003). Items consistently yielded the strongest pattern coefficients for their hypothesized factors, an encouraging result indicating that the factor structure replicated successfully in a second sample (Osborne & Fitzpatrick, 2012). Although the scale met the relative criteria for pattern coefficients, some coefficients were lower than the desired threshold of .40. Communalities tended to be strongest for Value, with one exception (Item V6), indicating that responses to items on this scale are most strongly associated with the underlying construct. Just under half of the variance in survey responses was explained by the three-factor structure (42%). The correlations were as follows: Value and Struggle, .33; Value and Growth, .32; Struggle and Growth, .19.
Standardized Factor Pattern Coefficients, Structure Coefficients, and Communalities for the Three-Factor Principal Axis Factor Analysis of the Mathematical Resilience Scale, EFA2.
Note. EFA = exploratory factor analysis. Pattern coefficients given in boldface have values of .40 or greater and signify items loading primarily with that factor. Pattern coefficients less than .10 were suppressed. The communality of measured variables is given by h2.
Item was reworded in EFA2. bItem is new in EFA2.
Internal Consistency Reliability Analyses for EFA2 and Resulting Decisions
Value scale
Nine items were included in the analyses, all of which were retained after factor analysis, yielding a very reliable scale (coefficient α = .90, 95% CI = [.88, .92]) with generally moderate interitem correlations (M = .51, minimum = .19, maximum = .81, SD = .17). Once again, interitem correlations were high between Items V1, V2, V4, and V5.
Struggle scale
The Struggle scale was revised using EFA1 results. Two new items were added (Items S9 and S10), and the wording was revised for three items (Items S3, S7, and S8). The three revised items were retained in the scale, with pattern coefficients greater than .40 (Item S3) or very close to .40 (Item S8). The two new items (Items S9 and S10) were also retained although the pattern coefficient for Item S9 was low on Struggle (.31), possibly because of multiple negative references that may have caused confusion (e.g., “When someone struggles in math, it doesn’t meant they have done something wrong”). Nonetheless, because the item contained content not reflected in any other item—that is, not attributing the cause of struggling to something done wrong—it was retained for the CFA.
The factor analysis extracted 10 items on a single factor with moderate reliability (coefficient α = .79, 95% CI = [.75, .82]) and small to moderate correlations between items (M = .28, minimum = .04, maximum = .50, SD = .10). The lowest interitem coefficient was between Items S10 and S9 (.04).
Growth scale
The hypothesized Growth factor contained eight items, and the factor analysis extracted all eight items on one factor with acceptable reliability (coefficient α = .82, 95% CI = [.79, .85]). Items were moderately correlated with each other throughout the scale (M = .37, minimum = .21, maximum = .58, SD = .09).
Summary: EFA2
All the items tested in EFA2 were retained for the CFA; however, items with low coefficients in EFA1 and EFA2 received closer scrutiny in the CFA step to determine whether they should be retained or removed to improve the theoretical meaning and fit of the final model. The resultant model from EFA2 contained three factors with 27 items, 9 on Value, 8 on Growth, and 10 on Struggle.
Study 3: CFA
Using Amos 18.0.0 (Arbuckle, 2009; Thompson, 2010) a CFA was completed on the third sample using the same items and factor structure extracted from the EFA2 study. The CFA technique uses maximum likelihood estimation to test the measurement model with the assumption that each item is an indicator of only one factor and error variances are uncorrelated (Kline, 2011). When the model fits well, relatively high standardized regression weights provide evidence of convergent validity, and relatively low correlations between the factors provide evidence of discriminant validity (Kline, 2011). The chi-square is not an optimal measure of model fit for a CFA because it is commonly significant when testing large samples, so alternate fit indices were used to measure degree of fit. The Tucker–Lewis Index (TLI) and comparative fit index (CFI) values preferably greater than .95 with greater than .90 an indication of acceptable fit, and root mean square error of approximation (RMSEA) values less than .06 were used as indicators model fit (Hu & Bentler, 1999; Netemeyer, Bearden, & Sharma, 2003; Yuan, 2005). In addition, the standardized root mean square residual (SRMR) less than .08 indicates acceptable model fit (Hu & Bentler, 1999).
Descriptive statistics were run after reverse-scoring Items G3 to G7 and Item G9 to ease interpretation. The mean item scores ranged from 3.74 to 6.07 (SD between 1.02 and 1.72), indicating a reasonable range and good variation.
Measurement Model Results
The CFA was completed to test the fit of the measurement model defined as Items V1 through V9 on Value, Items S1 through S10 on Struggle, and Items G1 through G7 and Item G9 on Growth. Path coefficients were all statistically significant (p < .05), but the model fit statistics indicated room for improvement: χ2(322, N = 290) = 737.6, p < .001, RMSEA = .07, 90% CI = [.06, .07], SRMR = .07, CFI = .88, and TLI = .87. Steps to improve model fit included an examination of the regression weights, modification indices, item content, and residual matrix as well as a closer look at items that were problematic in earlier steps. Revised model fit indices for each step are provided in Table 6. For Growth, Items G1 through G8 had satisfactory standardized regression weights of .54 to .73, but the error variance for Item G1 had a large modification index with both the Value factor and the Struggle factor. Combining this result with earlier evidence of a large secondary regression weight on Value for Item G1 in EFA1 caused us to be concerned that Item G1 may be multidimensional. This problem suggests that there may be differences between responses to the item wording that suggests anyone can learn math versus Item G1 (“Everyone can get better at math if they try”). Item G1 was removed from the instrument resulting in improved model fit. For the Value factor, Items V1 through V9 all had standardized regression weights of .65 or higher except Item V6, which had a standardized regression coefficient of .28. The wording of Item V6 was different in that it asked the respondent to consider the importance of mathematics to other people, whereas the remaining items asked the respondent to consider the importance on a personal level. Based on the wording and low regression weights for all three models, it was removed resulting in improved model fit. For the Struggle factor, Items S7 and S9 had low standardized regression coefficients from .34 for Item S7 (“People who are good at math may fail a hard math test”) to .36 for Item S9 (“When someone struggles in math, it doesn’t mean they have done something wrong”). These items contained wording that combined positive and negative references, and this may have confused respondents. Item S9 was also an item with low pattern coefficient in EFA2. When Item S7 was removed, the model improved, but when Item S9 was also removed it did not improve the model. In addition, when Item S7 was retained and Item S9 removed, this also did not improve model fit over the model with Items S7 and S9. The selected best fitting model, shown in Table 6 as Model 4, contained all items except Items G1, V6, and S7.
Model Fit Indices.
Note. df = degrees of freedom; CFI = comparative fit index; TLI = Tucker–Lewis index; RMSEA = root mean square error of approximation; RMSEA 90% CI = confidence interval for RMSEA; SRMR = standardized root mean square residual; AIC = Akaike information criterion; BIC = Bayesian information criterion.
Final Measurement Model
The final measurement model contained 24 items. Using criteria from Hu and Bentler (1999) model fit was deemed adequate using the criteria of RMSEA of .06 or less and SRMR of .08 or less, but below target for CFI and TLI: χ2(250,N = 290) = 535.9, p < .001, RMSEA = .06, 90% CI = [.06, .07], SRMR = .07, CFI = .91, and TLI = .91. The results were sufficient to support the three-correlated-factor model for the MRS. The covariances between the factors are all statistically significant (p < .05), and the correlation between the factors Value and Growth, Growth and Struggle, and Value and Struggle are .43, .24, and .40. Although these small to moderate statistically significant correlations between the factors provided evidence of discriminant validity, the correlation between Growth and Struggle was lower than expected. The internal consistency reliability was sufficiently high for affective survey research based on coefficient alpha (Comrey, 1988), .94, 95% CI [.93, .95] for Value, .73, 95% CI = [.68, .77] for Struggle, and .83, 95% CI = [.80, .86] for Growth, but there is need for improvement in the Struggle scale. Standardized pattern coefficients, which were all statistically significant (p < .05), are presented in the path diagram in Figure 1. The standardized and unstandardized regression coefficients are provided in Table 7.
Confirmatory factor analysis path diagram.
CFA Standardized and Unstandardized Regression Weights.
Subscale Differences
After analyzing response patterns using exploratory and confirmatory factor analyses, differences in trait levels were examined across demographic groups to provide further validation evidence (McCoach et al., 2013). Specifically, this portion of the study investigated whether resilient outcomes in mathematics are positively associated with higher levels in the Value, Struggle, and Growth measured in the MRS. Using the combined sample from EFA2 and the CFA (n = 579), student mean subscale scores were compared on the demographic traits of gender, race (White vs. all other), major (STEM vs. all other), highest level of mathematics coursework (calculus or higher vs. lower than calculus) and above or below average SRKM. Students who favor mathematics as indicated by taking calculus or higher, choosing a STEM-related career, or rating higher than average in SRKM were hypothesized to have higher scores on the three subscales of the MRS. In addition, the literature supported a hypothesis that the MRS scores would not vary across race or gender. These conjectures were based on findings from a meta-analysis by Ma and Kishor (1997) that suggested that gender differences in mathematics attitudes are either small or declining across time. A more recent study by Else-Quest, Mineo, and Higgins (2013) found gender differences in math self-concept, and expectations for success, but no differences in math value. Their work also suggests that gender differences in attitudes may weaken with inclusion of a control variable that indicates a favorable attitude toward mathematics such as the choice of a STEM major or high mathematics self-efficacy.
Descriptive statistics for the mean subscale scores for Value, Struggle, and Growth are shown in Table 8. A series of single-factor multivariate analysis of variance (MANOVA) were performed to test statistically significant differences between groups for a linear combination of the dependent variables Value, Struggle, and Growth. MANOVA was selected to test the multivariate effect of the combination score and to help control against an inflated Type I error rate due to correlated variables (Tabachnick & Fidell, 1996).
Scale Means and Standard Deviations.
MANOVA Results
The results revealed that students whose SRKM was higher than average were significantly different from those with lower than average SRKM, Pillai’s trace = .16, F(3, 575) = 36.97, p < .01, partial η2 = .16. In addition, students who have taken calculus or above were significantly different in the multivariate score than those who have not taken calculus, Pillai’s trace = .05, F(3, 575) = 9.95, p < .01, partial η2 = .05. Students also differed by STEM major, Pillai’s trace = .07, F(3, 575) = 14.34, p < .01, partial η2 = .07, and by race, Pillai’s trace = .01, F(3, 575) = 2.82, p < .05, partial η2 = .01. No statistically significant differences were found by gender.
ANOVA Results
A series of follow-up ANOVA were performed to further analyze the significant results. Students in STEM majors scored significantly higher than non-STEM in the Value subscale, F(1, 577) = 38.33, p < .01, d = 0.73, and the Struggle subscale F(1, 577) = 5.42, p < .01, d = 0.27. Students with higher than average SRKM scored significantly higher than those with lower SRKM in both the subscales of Value F(1, 577) = 98.23, p < .01, d = 0.84, and the Growth F(1, 577) = 34.73, p < .01, d = 0.50. Students who have taken calculus or higher scored significantly higher than those who have not taken calculus on the Growth subscale, F(1, 577) = 6.63, p < .01, d = 0.47, and Value subscale, F(1, 577) = 28.88, p < .01, d = 0.22. There were no significant main effects for race for any of the three subscales.
These results are generally favorable when compared with hypotheses. Evidence of construct validity is provided by significant differences in the MANOVA for students who favor mathematics as indicated by taking calculus or higher, choosing a STEM-related career, or rating higher than average in SRKM supporting the ability of the MRS to differentiate between students who favor mathematics versus those who do not. Although contrary to the hypothesis, the MANOVA also indicated significant differences based on race, but the small effect size and lack of significant follow-up ANOVA on the subscales reduced the impact of this result.
Discussion
This research developed an instrument that can be used to distinguish students who may be more likely to persist in the study of mathematics when they face setbacks from those who are not likely to persist. Recently there has been a heightened interest in ways to improve participation and persistence of students in STEM majors in universities. This instrument may be used by researchers in motivation and mathematics education and by policy leaders seeking ways to improve representation in STEM careers for all students and a better understanding of reasons for attrition from STEM careers. It will be useful for counselors interested in identifying and helping students who are mathematically talented, but whose affective traits make it less likely for them to persevere in their studies.
The evidence provided in this study supports the reliability, content, and construct validity of the MRS as a measure of a construct defined by three correlated factors: Value, Struggle, and Growth. The factor labeled Value addresses the belief that math is essential for future success. The factor labeled Struggle gauges respondents’ belief that experiencing challenges and difficulties is a normal part of working on math. Finally, Growth refers to the belief that math can be learned by anyone and is not limited to those with a predisposition or “math gene.” EFA1 included a Resilience factor, but the items in EFA1 were multidimensional, indicating that the attitude represented by responding with grit, stamina, or determination to persevere when facing significant challenges is captured at least in part by the three remaining scales. Ultimately, it was decided that a self-report instrument that measures whether the subject believes he or she will respond to difficulties in mathematics with grit was less useful than finding attitudes that are predictive of outcomes indicating Resilience. Based on theory and the EFA1 results, the items measuring student assessment of response to difficulties in mathematics were removed from the instrument. The measurement model based on the CFA did not provide the level of fit to meet the criteria suggested by Hu and Bentler (1999). Future research should explore ways to revise some of the items that may ultimately improve model fit.
The analyses of differences in subgroups supported the validity of the use of scale scores in research. The students who have taken more advanced mathematics, have chosen a STEM major, or have stronger self-efficacy also have higher scores on the MRS as measured by the MANOVA as well as on the subscale scores. The results are supported by other results reported in research literature. For example, in a meta-analysis of the relationship between attitudes toward mathematics and achievement in mathematics, Ma and Kishor (1997) found a significant but not strong relationship between attitudes toward mathematics and achievement (δ = 0.12). In addition, the lack of a significant difference in the MRS scores due to gender provides additional validity evidence, contributing to the strength of the instrument. The constructs of Value, Struggle, and Growth are able to differentiate based on malleable attitudes predictive of persistence in mathematics in spite of setbacks but not based on gender, which is a desirable quality in affective instruments.
The Struggle subscale represents a new approach to the study of attitudes toward mathematics. The evidence gathered in this study suggests that Struggle contributes an important dimension in identifying attributes that drive a student to persist in mathematics when faced with difficulties. The Struggle subscale score for STEM majors was significantly higher than non-STEM majors. However, the Struggle subscale scores were not as internally consistent as the other subscales (.73 in CFA) and the correlation between Struggle and Growth was small (.24 in CFA), indicating a low-level association between the belief that people can develop mathematical ability and the belief that struggle is a normal part of learning mathematics. Future research on this subscale should include qualitative research to fine-tune the items to better describe the Struggle dimension. For example, the original items were worded to assess attitudes toward the normalcy of Struggle in others who are studying mathematics, but perhaps revising this to a self-focused assessment may improve the consistency and meaningfulness of the scale.
The results for the Value subscale provide strong evidence of the reliability and validity of the Value scores as a measure of student belief that mathematics is valuable for future goals. Coefficient alpha for the Value scores of at least .90 for all three studies indicates that the scores on the items are internally consistent. A comparison of scale scores indicated that student scores on Value did not vary significantly based on gender or race, which aligned with the hypotheses. Value scores were higher for STEM majors, for those with high SRKM, and for those who have taken calculus or above.
Similarly for Growth, the subscale was shown to be internally consistent in all three studies, with a coefficient alpha of greater than .80. These results also provide evidence of validity in significantly higher scores on the Growth subscale for students who assessed themselves with a high SRKM and for students who have taken calculus or higher.
Although evidence of reliability and validity for each of the subscales is important in establishing the reliability and validity of the instrument, the results of the MANOVA provided additional validity evidence. Students who favor mathematics as indicated by selecting a STEM major, taking calculus or higher, or having a higher than average SRKM scored significantly different from their counterparts on the MRS. Although scores were significantly different by race for White versus all other, the effect size and lack of significant follow-up ANOVA reduced concern over this result. This evidence supports the claim that the instrument can distinguish students who are more likely to persist in the study of mathematics from those who are not.
In future research this instrument can be used to examine the impact of educational interventions on noncognitive traits, and in particular the impact of interventions on motivational factors. Researchers might also test whether Value, Struggle, and Growth predict resilience in mathematics, which might be gauged by examining whether students who academically struggle on an assignment or throughout a semester, persist with their studies. In addition, studies of the relationship between the MRS and other affective variables such as mathematical anxiety, mathematical self-efficacy, and other motivation variables such as those reported in Eccles (1983) would be informative on two fronts. They would provide an opportunity to study the convergent and discriminant validity of the MRS and would also expand the knowledge based on the determinants of and relationships between different attitudes toward mathematics.
Limitations of These Studies
These results had a number of limitations in the realms of generalizability, the strength of pattern coefficients, and measurement model fit. Although the members of the sample used for EFA1 came from multiple institutions of higher education, it contained a large proportion of respondents who were professional insurance actuaries, and hence limit the generalizability of results. Nonetheless, the initial sample provided evidence to support the validity of the factor structure in a sample of actuaries drawn from a population that is hypothesized to have mathematical resilience based on persistence in studying mathematics. The samples used in EFA2 and the CFA were more representative of the target population, but the students attended a single university and included only students in one of eight large lecture hall classes. Furthermore, the MRS has only been validated on college and university students and adults, limiting its current applicability to students entering college or older. Nonetheless, the theory developed herein may, with small adjustments, be just as relevant for younger students to persist in the study of mathematics in spite of difficulties. An area of future research would include conducting a validity study on students of other age groups. Finally, the fit of the measurement model based on the CFA was slightly below the criteria, suggesting the need for future research to explore ways to revise some of the items that may ultimately improve model fit.
Although this study provides a theoretical basis for the connection between these three factors and outcomes that are indicative of mathematical resilience, the number of outcomes available to analyze as evidence of external validity in this study was limited. Future research in this area could lead to improvements in understanding of why some students continue to study mathematics while others do not, a key component toward the goal of increasing representation and participation in STEM majors.
Considerations for Practice
The current study of the MRS provides counselors with a new approach for helping students who experience challenges studying mathematics and their teachers. Counselors may find the MRS to be a useful instrument to identify reasons why students lack persistence in mathematics. Counselors can use this instrument to identify students who score low on the MRS and provide them with information and interventions to improve their resilience. These approaches are beneficial for students at all levels of mathematics. For example, when students are enrolled in lower level mathematics classes, they may not realize that students at all levels of mathematics struggle with the content at some point. When they realize that everyone struggles, they may be more likely to persist and less likely to develop anxiety over what they perceive as their own limitations. Interventions to encourage a growth theory of learning can also help students recognize that with effort, they can improve. Students for whom mathematics has not been a challenge will likely face a point in time when it becomes difficult for them. Counselors can use theory on the constructs in this instrument to provide perspective to the student, reducing frustration and anxiety, and improving motivation to continue. Counselors can also share this information with teachers when working with groups or individual students. Teachers can help students by sharing their own beliefs on the usefulness of mathematics and their experiences when they struggled but ultimately succeeded through hard work.
Footnotes
Declaration of Conflicting Interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
The author(s) received no financial support for the research, authorship, and/or publication of this article.