More than Motivation: The Combined Effects of Critical Motivational Variables on Middle School Mathematics Achievement Free

The importance of motivation in shaping the mathematical experiences of children has been well established for some time. Over the years researchers have focused on students’ feelings about their own success in mathematics, their ability beliefs and self-efficacy, aspects of tasks that foster intrinsic and extrinsic motivation, and on the social nature of motivations in context (Middleton & Jansen, 2011; Middleton & Spanias, 1999). Each of these theoretical perspectives¹ has characterized the beliefs, values and goals of students trying to navigate the educational landscape, and the effects of these beliefs, values and goals on their decisions to put forth effort, persist in difficult tasks, and on their choices of academic and vocational pursuits.

In the grand scheme of things, it can be said that, along with mathematical knowledge, students’ motivations are the most direct predictors of student performance and achievement in which teaching, learning, and school organization can have an impact (Eccles & Wigfield, 2002). This is due to the fact that they are themselves influenced by a variety of these psychological, social, and cultural determinants, which interact in the life of the learner. It can be argued, therefore, that motivations serve as a lynchpin, connecting the psychosocial baggage carried by the learner, with the instructional environment, enabling to a great extent the potential future directions that he or she may take (Husman & Lens, 1999).

The purpose of the study reported in this manuscript was to test a model of this interaction, utilizing a small set of important motivational factors that have been found to be influential in the literature on student academic performance. Instead of exploring their impact individually, and separately, I analyze their mutual influence using structural equations modeling, a statistical technique that allows for analyzing the relationships among variables, some of which display direct effects one upon the other, and some of which show indirect and combined effects. The logic of, and problems associated with, these techniques are reviewed later in the Methods section. For the present, I review the theory behind each of the variables shown to impact student learning and achievement, arguing how they constitute a single hypothetical structure that as a composite provides explanatory power for students’ mathematical decisions and subsequent performance. In the present study, partly due to the nature of the data set employed (the High School Longitudinal Study of 2009), I focus on middle-level motivations—those that relate to (1) the utility of courses and course-taking, (2) interest in mathematical pursuits as a general orientation, (3) mathematics self-efficacy, (4) the degree to which students maintain an identity as a “mathematics person,” and (5) the effort students are willing to expend in the service of mathematical performance. Each of these motivational domains is reviewed separately, but I attempt to show that they each support the other in creating a reasonably robustattitude in students that assists them in predicting the potential value of mathematical engagement.

One of the most well researched, and most intriguing domains in mathematics motivation for its perplexing findings concerns the role of interest. Although interest has been defined in different ways operationally, a good working definition of interest was promoted by Dewey (1913) as involving a connection or identification of one’s self, with some object of the interest (some person, thing, or action), because that object is critical for the person to engage in some self-initiated activity. In other words, interest is the direction of attention and emotion toward any thing that can help us do what we want to do. As such, it involves elements of curiosity, goal-directedness, and personal identity (Middleton, Lesh, & Heger, 2002).

In the psychological literature, interest is posited to possess both state and trait characteristics. As a state characteristic, any task we set ourselves to has what is often called “interestingness.” Many researchers characterize this interestingness as a function of the environment—some environments because they emphasize novelty, challenge, social interaction or other hypothesized characteristics, have a higher probability of engendering interest in students than others. Situational interest, therefore, ebbs and flows in mathematical tasks as students face challenges, encounter novel or familiar content and contexts, and as the motivational climate in the classroom affords excitement, anxiety or boredom as the case may be (Schweinle, Meyer, & Turner, 2006). Key to my point here is that, because much of students’ motivation in mathematics is due to the way in which we design mathematical tasks, tools, and classrooms, careful attention to the redesign of these factors may afford the development of situational interest in students—thus engendering more excitement, on-task behaviors, and subsequent mathematical learning (Middleton, Leavy, & Leader, in press).

Over time, because tasks within mathematics courses, and because mathematical classrooms in a school or in a community, tend to show overall similar characteristics, situational interest (or lack thereof) will begin to stabilize around particular kinds of tasks and pursuits. It does not take too long in the U.S. education system for students to begin to seem themselves as a “math person” or “not a math person” because of relatively stable patterns of success or failure, or enjoyment or boredom, over the course of their learning experiences. These longer term attitudes toward academic subject matter are called “individual interests” or “personal interests.” Hidi and Baird (1988) characterize personal interests as an attitude students bring to the learning situation. For example, students typically come to a middle school mathematics classroom with preconceived notions of what they will expect, based on what they have experienced in the past. A major orienting factor for their learning behavior, therefore, is their interest (or disinterest) in mathematics that has resulted from these prior experiences.

Key to the importance of personal interests for mathematics instruction is the fact that any individual will come to class with her or his own set of interests, some of which are mathematical and some of which are not. The more classroom tasks tap in to these personal interests, the more effort and persistence the student is willing to expend to achieve the goals of the task (e.g., Hidi, 1990; Hidi & Baird, 1988; Krapp, 2000; Renninger, 2000; Schiefele, 1991).

Putting these ideas together, we can characterize situational mathematical interest as a function of the goals, values, and desires the student brings into the realities of the learning context in the moment, and personal interest as the accumulation and organization of these moments in a students’ memory (Middleton & Jansen, 2011). An important implication of this is that interest must be studied as a content-specific construct. No person is genuinely interested in every thing, but only in some subset of things depending on their histories and circumstances. We see for mathematics, people highly interested who are less interested in other subject matter, and we find for students interested in some subject matter, disinterest in mathematics (Mitchell, 1993). These personal interests become more and more stable over time—consolidating into an academic identity where subject matter plays a key role (I will review the literature on identity shortly). Moreover the degree of agreement between interests and achievement in subject matter, especially mathematics, becomes closer and closer over time (Schiefele, Krapp, & Winteler, 1992), indicating that, the more a person achieves, the more interested in the subject he or she becomes. Likewise, the more interested a person becomes, the more one achieves. The converse of these statements is also, unfortunately, true.

The construct of interest is aligned with values, enjoyment, intrinsic motivation, identity and self-efficacy in mathematics (Fredricks & Eccles, 2002; Middleton & Toluk, 1999), but is separable empirically, from those constructs (Schiefele et al., 1992). Emotionally, when one experiences pleasure or excitement, this serves both an informational role to alert the student to the fact that the situation is likely to become interesting, and to provide reinforcement contingencies—rewards and punishments—that induce the student to persist or desist in their behavior. These emotional reactions are dependent upon the personal significance of the task and its outcomes, as well as physical and social factors (Krapp, 2000; Renninger, 2000; Schiefele, 1991; Wigfield & Eccles, 1992, see also Goldin, 2002 for his lovely discussion of affective structures). Likewise, when one has the opportunity to engage in an interest, he or she can anticipate that the activity will engender these positive feelings.

Research over the past 20 years has shown, time and time again, that interest significantly, positively predicts achievement outcomes in mathematics, as well as other academic subject matter. Trends in motivation and achievement in the middle grades have repeatedly shown that mathematics interest begins rather high, but is reduced steadily over time to reach its nadir early in the high school years), roughly paralleling achievement (Eccles & Wigfield, 2002; Fredricks & Eccles, 2002; Frenzel, Goetz, Pekrun, & Watt, 2010; Jacobs, Lanza, Osgood, Koller, Baumert, & Schnabel, 2001; Watt, 2004).

Yet despite all we know about the importance of Interests in predicting achievement, the strength of the empirical relationship between the two is not as strong as may be surmised. Schiefele, Krapp, and Winteler (1992), for example, performed a meta-analysis on 31 studies of interest effects on academic performance, and found an overall correlation between the two of only .31. Mathematics, when analyzed separately across 16 studies, showed a correlation of .32. Thus, only 10% of the overall variance in mathematics achievement scores could be explained by differing levels of interest in students.

Likewise, Middleton, Leavy and Leader (2013), analyzed the impact of a variety of motivation variables, including utility, interest, attributions and challenge on the mathematics achievement of middle grades students. In this study, though significant statistically, the combined affect of the variables accounted for only about 6% of the overall variation. Clearly something else is going on to translate students’ motivations into patterns of behavior that ultimately determine their achievement patterns. I now turn to another candidate, the notion of mathematics identity, and its influence on achievement.

Identity is a problematic construct in mathematics education because it means different things to different people. To some, identity is belief, cognitive in nature, held by a particular person, directed toward his or her self-image as a mathematically competent learner (see Grootenboer, Smith, & Lowrie, 2006, for an excellent discussion of the different perspectives of identity development). Under this perspective, identity is a self-definition, individual in nature, often tied to the person’s attributions, efficacy expectations, and mindset (e.g., Oyserman, Elmore, & Smith, 2012). To others (e.g., Boaler & Greeno, 2000; Holland, Skinner, & Cain, 1998; Martin, 2007), identity is the place one occupies and the role one sees themself playing out, in the social space of mathematics classrooms. Under this perspective, the give and take of tasks in the social context defines whether the student considers him- or her-self as fitting in socially. Even more radically (e.g., Wenger, 1999), identity is defined as the place the person occupies as seen by the collective itself. These theoretical perspectives are each useful, and we can see that they form a kind of spectrum ranging from individual/cognitive to reflexive/situative to collective/anthropological treatments of the relationship held by the individual to the field of study, here mathematics. Because the first perspective (cognitive) overlaps significantly with the concepts of interest and selfefficacy, two of the other constructs studied in this paper, I have chosen to focus this discussion on identity especially on the situative perspective.

It is clear from the research on mathematical attitudes, emotions, and motivation (see reviews by McLeod, 1992; Middleton & Spanias, 1999) that children develop affinities for and against the school mathematics practices in which they are engaged. These mathematical identities develop in a network of nascent identities that grow from peer interactions, cultural identification, and their interests and predilections in a variety of roles and subjectmatter (Lemke, 1997).

In mathematics, Cobb, Gresalfi, and Hodge (2009) discuss two definitions of identity that differ along the individual versus social dimension alluded to earlier in this section: Personal versus Normative Identity. Students’ personal identities with respect to mathematics involve their understandings of what “ought” to be done, who “ought” to do it, and the value of they themselves doing it. As such, it overlaps significantly with personal interest as a motivational domain (Middleton et al., 2003). Additionally, because such “oughtness” is tied to students’ beliefs about their mathematical capabilities, personal identity is rife with selfefficacy statements—identity turns the evaluation of personal success and failure in mathematics into a statement about who one is and the role of mathematics in defining one’s self.

Normative identity involves three distinct sets of classroom norms that constrain students’ engagement toward collectively fruitful ends.² These norms include the ways in which students can express agency (e.g., through volunteering answers, helping others, or making a joke), the persons to which student behavior is accountable (typically other students and the teacher, but parents and administration or extraclass peers may also be included here if authority lies at least partly within them), and what counts as competence mathematically. Agency, accountability and competence thus define the range of potential roles students in the classroom can play. Students who have some sense of autonomy, who can self-evaluate and coordinate those evaluations with the teacher, and who display mathematical competence in a manner that is deemed “appropriate” will have normative identities that are productive for the class goals, while those who display less autonomy, who require others to tell them when their mathematical behaviors are appropriate or not, and whose competence falls outside the acceptable boundaries will exhibit normative identities that are un- or less-productive for the class goals. In other words, students whose normative identity is positive tend to become more central to the organization and conduct of the class, while students whose identity is negative tend to play the role of outsiders (Gresalfi, Martin, Hand, & Greeno, 2008).

Consonant with the description of personal identity and personal interest I have provided in this paper, there has been a long tradition of examining students efficacy beliefs and their impact on students’ learning and achievement in mathematics (e.g., Bandura, 1982; Pajares & Miller, 1995; Zimmerman, 2000). Mathematical self-efficacy concerns the personal judgments people make about the likelihood they will be successful (or not) when engaged in mathematical tasks. A combination of factors work together to form students’ mathematical self-efficacy (Urdan & Schoenfelder, 2006). Briefly, the cumulative successes and nonsuccesses students experience in a domain are evaluated, and the causes of these successes and nonsuccesses lead students to categorize themselves as successful or nonsuccessful.³ In addition to personal judgments of efficacy, students also make comparisons of their successes and failures to those of others, and make judgments about their own success relative to the social norm. They emulate the strategies of others and coordinate their values with those of the class, including what we might find to be nonproductive strategies and values in terms of learning mathematical content. Lastly, verbal persuasion from peers, the teacher, parents and others in authority are used by students to determine the worth of their successes and failures.⁴

Positive self efficacy is strongly and positively related to mathematics achievement (Skaalvik & Skaalvik, 2004). Students who have historically been successful in reasonably difficult mathematics tend to develop positive self-efficacy. They tend to see mathematics more as a personal interest, and they tend to develop positive personal and normative identities with respect to mathematics. They tend to feel autonomous and independent in completing tasks, and they tend to have a supportive teacher (Goodenow, 1993; see also McCaslin, 2009). Each of these factors are intertwined, contributing to a positive sense of well-being, even a positive anticipation for engagement in mathematics. It is the case, however, that the majority of students tend to develop less and less self-confidence—lower self-efficacy—as they move through the middle grades and into high school and higher education (Meece, Wigfield, & Eccles, 1990; Hackett & Betz, 1989, 2006).

Husman, Pitt Derryberry, Crowson, and Lomax (2004) discuss utility, or perceived instrumentality of classroom asks as a future-oriented belief about the value a task or pursuit has for the learner. In essence, we perceive tasks, tools or behaviors to be useful when they are instrumental in achieving some envisioned future goal. Utility is thus tied to a student’ interests and identity in that it is framed by their thoughts about their futures as well as their hopes and fears about the present. Utility can be seen as a mediating variable in this context—the perceived utility of a task or pursuit makes the pursuit more interesting (situational interest), and if a field of study is seen as having use for the learner, it increases the probability of the subject matter being perceived as interesting (personal interest). Related to achievement, perceptions of utility lead students to develop mastery goals and are tied to a student’s perceptions of competence: The more competent a person thinks he or she is, the more likely they are to see mathematics as relevant and useful (Chouinard, Karsenti & Roy, 2007). It has been consistently shown that many future-oriented goals students create (for example occupational aspirations, future education pathways, etc.) are instrumental goals, parlaying students’ perceptions of utility into plans of action to achieve these instrumental goals (Lens, Paixao, & Herrera, 2009; Nuttin & Lens, 1985). Students show strong identity related to such goals, and work to achieve them if they feel they are attainable.

Again, like interest, identity, and self-concept, we see a sharp decrease in students’ feelings about the usefulness of mathematics as they progress through the middle grades, through high school, and into college (Hacket & Betz, 1989; Chouinard & Roy, 2008; Hoffmann, Krapp, Renninger, & Baumert, 1998; Kessels & Hannover, 2007).

Effort is a critical intervening variable in the prediction of achievement from students’ motivations. High motivation, especially intrinsic motivation or perceptions of instrumentality of the task tend to engender moderate to high effort, which in turn, tends to result in greater performance in mathematics. This success, in its own right tends to yield greater mathematical self-efficacy, conscientiousness in work, thus making effort more effective (Vasalampi, Nurmi, Jokisaari, & Salmela-Aro, 2012; Yunus, Suraya, & Wan Ali, 2009). Effort, therefore, is the degree to which motivation becomes translated into learning behaviors.

In the attribution literature, effort (or, more specifically, lack of effort) can be seen as the most productive reason for failure a person can attribute. This is because with lack of effort, there is a clear behavioral alternative to follow if the student wants to become successful— expend more effort! Throndsen (2011), for example, found that high mathematics achievers tend to attribute lack of effort in explaining why they sometimes fail, while low achievers tend to focus on lack of ability as the cause of their failures.

For success attributions, it is a bit more tricky. In general a student needs to feel that their successes are attributable to moderate to high ability and applied effort. This enhances mathematical self-concept, leading to increased identity with mathematics, and the positive benefits of the application of effort— increased success in mathematics class. There are important gender differences for science, technology, engineering, and mathematics (STEM)-oriented students, however. It has been shown, for example, that women who think they work harder than others to succeed in STEM pursuits tend to feel less of a sense of belonging in the field, and a decreased motivation for the subject matter (Smith, Lewis, Hawthorne, & Hodges, 2013).

The work of Carole Dweck (Dweck & Leggett, 1988; Good, Rattan, & Dweck, 2012) shows that students who believe ability is a result of effort tend to have an incremental view about intelligence. These students tend to believe that, even if they are smart, they can get smarter by expending more effort in a pursuit. Other students hold a fixed view about intelligence. They tend to believe that when they fail, there is nothing to be done, because they don’t possess the cognitive capacities to do well, and tend to opt out. These beliefs play out in the ways in which effort is expended, with students holding incremental views expending more effort, and more efficient strategies than students holding fixed views of intelligence. Of course, this plays into students’ mathematics identities and self-efficacy beliefs (Schunk, Pintrich, & Meece, 2010).

In the present study, effort is considered a complex variable consisting of both time expended on mathematical pursuits (e.g., homework), and the diligence students display in coming to class, being prepared with materials and readings, and the like. Effort, after all, isn’t just intensity of work, but intensity of work applied over some length of time.

The impact of each of these motivational variables on achievement has been demonstrated in a variety of studies ranging from qualitative descriptions of classroom interactions (e.g., Middleton, 1995, 1999; Turner & Patrick, 2004) to relatively large-scale interventions (e.g., Mitchell, 1993; Middleton et al., in press). The general consensus of these studies paints a poor, but not hopeless picture for mathematics instruction. Typically students do not see the content of mathematics as a personal interest. They do not often report situational interest in mathematics tasks. They do not see it as being useful in a personal sense, and they tend to show decreasing selfefficacy and identity with mathematics.

But this is not the case with all students. Many do identify as a “mathematics person,” showing interest and efficacy in their pursuit of mathematically sophisticated professions. This variability in students’ beliefs and their relation to student achievement in mathematics has the potential, if mined carefully, to help us develop better and more personally relevant curriculum, tools, and pedagogical routines (e.g., Middleton et al., in press).

What we do know about these variables is that they are not all the same thing. Each contributes some unique variance to a student’s mathematics achievement. It has been shown that situational interest and personal interest are separable, as well as utility (meaningfulness) and involvement (akin to identity) (Middleton et al., in press; Mitchell, 1993). Selfconcept, a construct overlapping significantly with self-efficacy shows positive effects on personal interest, but the reverse is not as strong (e.g., Marsh, Trautwein, Lüdtke, Köller, & Baumert, 2005).

As can be discerned in my brief descriptions of these motivational variables, the impact of any one variable does not occur in isolation from the other variables. Feedback loops from mathematical experience augments or reduces students’ interest, which in turn impacts perceived utility, effort, and self-efficacy beliefs. Identity and self-efficacy, for their own part, seem to play off each other as students attempt to discern who they are in relation to the larger meaning of mathematics in schooling and future occupational aspirations. The remainder of this manuscript tests a model of this interaction developed initially by Middleton (1995) and Middleton and Toluk (1999), and recently revised by Middleton et al. (2013). I do not go into the previous model deeply, but steer the reader to these publications to see the research upon which the structural equations model tested in this article is based.

Serendipitously, the High School Longitudinal Study 2009 has just released its first administration of data. The present study provides the opportunity to study the results of the first 8 years of U.S. students’ mathematical motivations in their first year of high school before becoming acclimated to the secondary culture and routine of mathematics, and to link those motivations to achievement in algebraic concepts and processes. I expected to see the theoretical interactions among these motivational variables validated in this study, with the added power of a huge nationally representative sample of middle schoolers to allow predictions of effects across the network of interactions.

The High School Longitudinal Study of 2009 (HSLS:09, National Center for Education Statistics, 2011) is a long-term study of 21,000 ninth graders in 944 schools administered by the National Center for Education Statistics (National Center for Education Statistics, 2009). Data collected in the study focuses on students’ decisions on what courses to take, what occupations to pursue, their aspirations for higher education and their reasons for engaging in these pursuits. STEM pursuits, in particular, including mathematics and science courses, majors and careers, are a special focus of the study. Detailed demographic information and surveys of students’ parents, mathematics and science teachers, and school administrative and academic support personnel are also collected.

The initial administration of the HSLS:09 began in fall, 2009, surveying ninth graders on their past experiences in middle school, and on their current attitudes, motivation, and achievement. Students were surveyed again in the spring of 2012, and a follow up survey will be administered again in 2013 following expected graduation. Dropouts are also followed on this schedule. Future data collections are planned in 2016 and 2021 to learn about participants’ life trajectories and markers of success. This study reports only on the first administration.

HSLS:09 generated a nationally representative sample of 944 high schools, including both public and private schools. This makes the data set representative of communities in the United States. Following this initial sampling, approximately 25 ninth graders were selected from each school. The resulting sample totals approximately 24,000 students, representative of all ethnic categories and socioeconomic strata. Students’ parents, mathematics and science teachers, and other school personnel were then asked to complete surveys on their student’s behaviors and experiences. I only analyze student data at the national level in this study.

The HSLS:09 dataset provides several types of sampling weights to account for the complex survey design and produces estimates for the target population of choice. These weights allow for multilevel analyses across a variety of factors (e.g., students within ethnicities, teachers within schools, etc.). For this study the two pertinent subpopulations described are sex and Ethnicity. The appropriate variable weights provided by National Center for Education Statistics were applied to provide accurate descriptive statistics for these factors. The survey items and mathematics assessment items show minimal nonresponse bias. Some missing data, therefore, were imputed. For the present study, students’ sex and ethnicity have some imputed scores.

Eighteen items from HSLS:09 were utilized in the construction of motivation and achievement indices in the present study. Table 1 provides a listing of items for each hypothesized scale. HSLS has ready-built scales for some of the hypothesized variables (e.g., identity). Some hypothesized variables, however, were not constructed by the HSLS authors. For that reason, I developed scales by examining the wording of each of the mathematics related attitude items, and placing them into a scale that fit the hypothesized variables best theoretically. In doing so, I was able to eliminate items that proved to be unreliable, creating a more robust set of measures for the hypothesized motivation variables. For motivation variables, I eliminated cases with missing values listwise so that all modeling utilized real data. The overall nonresponse rate across all constructed scales reduced the overall numbers of student cases to 17,602 for analysis of mean responses for motivation scales.

The model developed by Middleton et al. (in press), and being validated in this present study is illustrated in Figure 1.

It is a drawback of secondary data analysis that not all variables used in one theory are operationally defined in exactly the same way by the authors of the second data set. What is lost in sensitivity is gained in robustness and confidence due to the large sample size in HSLS:09. The model utilized in the present study differs from the HSLS:09 variables in subtle ways, stimulation and control are variables assessing the degree to which mathematics is exciting (or boring), and under the personal ability of the student. Items in HSLS:09 dealing with these concepts, as predictors of Interest are combined into the Interest scale. Likewise, items measuring attributions and confidence are combined into the self-efficacy scale for the present study. Self-efficacy derives from the same root theory which assumes that people’s classification of their successes and failures in academic tasks, and their reasons for ascribing feelings of success and failures serve as a basis for generalized feelings of efficacy and control (Bandura, 1994; Weiner, 1979). Effort, utility and achievement remain unchanged in their definition. It is assumed that the differences in the operational definitions of these variables will add little spurious variation to the overall model.

Chronbach’s alpha was computed for each of the motivation subscales and for the overall set of motivation variables to determine the additivity of items within scales and as an overall index of motivation. In addition, the composite reliability (e.g., Raykov, 1997) was computed on the five motivation scales as indicated by their structure in the structural equation model. This procedure accounts for nonorthogonality of subscales, which is assumed in classical test theory. Table 2 lists these values. Results show that scales have moderate to very high internal consistency. Taken as a whole, the reliability of the constructed scales was determined to be adequate for the structural equations model.

Normal probability plots indicated that all variables were distributed normally, and that linear models are appropriate techniques for analyzing differences in groups and evaluating the structure of the variables in the factor and SEM analyses.

HSLS:09 includes a 40-item assessment of mathematics achievement. The test covers major concepts in high school algebra, including:

1. Proportional relationships and change;

2. Linear equations, inequalities, and functions;

3. Nonlinear equations, inequalities, and functions;

4. Systems of equations; and

5. Sequences and recursive relationships.

In addition, algebraic processes were measured, addressing the following:

1. Demonstrating algebraic skills

2. Using representations of algebraic ideas

3. Performing algebraic reasoning

4. Solving algebraic problems

The test was administered electronically, in two stages. In the first stage, all students took all items. The items in the second stage were assigned to students based on their performance in the first stage: low, moderate, or High. Thus, no students received all 72 possible items. They were limited to 40 during the 40-minute assessment period.

Results were subjected to an item response theory (IRT) 3 parameter logistic model. This model estimates the probability that a student will respond correctly to an item based on their responses to the other items. Therefore, a student’s probable number correct out of 72 items was estimated and recorded. The IRT estimate of the number of items each student would have scored correct was used as the measure of student performance in the study.

Table 3 shows means and standard deviations for motivation variables and mathematics achievement across sex and ethnicity. Results show significant differences for boys versus girls for all motivation variables (p < .001), with boys showing statistically higher motivation for mathematics across all categories except effort. Cohen’s d, a common statistic, scaled in standard deviation units, was used as a measure of effect size for each of the variables. Boys showed greater mathematical Identity (d = .10), and feelings of Utility, Self Efficacy, and Interest in mathematics than Girls (d = .05, .18, and .38, respectively). Girls, for their part, showed greater willingness to expend effort to become successful in mathematics-related activities (d = .20). There nonsignificant sex differences in mathematics achievement across the sample (see Table 3). Significant differences also existed for each of the motivation variables by ethnicity (all p<.01). Knowledge that they exist may assist future research in this area, so I include this fact. However, because these differences are subtle, and the root causes of these differences are not hypothesized in this study, I have chosen not to discuss them here.

Structural equations modeling is a set of techniques that use regression to determine the direction (sign) and magnitude (0-weights) of relationships among a set of variables. Items on HSLS:09 are considered indicator variables—pointing to an underlying latent variable (the motivation scales). When a structure among the latent variables is posited, the resulting matrix of intervariable statistical relationships can be used to test the direction, magnitude and overall goodness of fit of the posited cognitive relationships.

In the present study, partial least squares (PLS) regression was used to determine interitem relationships among indicator and latent variables in the structural equations model. PLS methods have advantages over strictly linear algorithms in that PLS can fit models with nonlinear relationships and non-Gaussian distributions among the variables in addition to the traditional linear and Gaussian situations. This results in less sensitivity to colinearity, and the ability to apply to nonlinear relationships among latent variables.

PLS is biased for small samples because the optimization is local rather than global; however, as sample size increases, PLS shows less bias. For the relatively small number of indicator variables (18), the small number of latent variables (6) and hypothesized paths (8), and extremely large sample size (17,874), PLS can be used to make inferences about parameters, namely path weights in the model. Bootstrap resampling was employed to generate sampling distributions for each of the latent variables so that standard error of measurements could be estimated. Because of the large sample size, 100 resamples sufficed to converge. Structural equations modeling was performed using WarpPLS 3.0 (Koch, 2012).

Figure 2 shows the hypothesized structure with directed links indicating the flow of causality in the model.

As can be seen, the direction and magnitude of the paths confirm the Model developed in Middleton et al. (in press). Note that the P weights from effort and interest to achievement are negative. This is because of higher levels of achievement corresponding to low values on the Likert items.

The model shows excellent fit indices. The average path coefficient is 0.276, p < .001. The average R-squared is 0.186, p <. 001, indicating that, overall, the predictor variables in the model account for around 19% of the variance of each sink. Moreover, the average variance inflation factor was 1.193. An average variance inflation factor less than 5 indicates low colinearity among predictor variables (Craney & Surles, 2002). The computed value is very low, indicating that latent variables are nearly orthogonal. Principal components analysis of the latent variables confirmed this near-orthogonality. Standard errors for path coefficients ranged from .006 to .009, suggesting that the large sample size generated a very narrow Bootstrap sampling distribution for the coefficients.

Overall, the direction and magnitude of the proposed paths among latent variables are as hypothesized. The use of such a large sample size was instrumental to adding confidence that the variables are separable and interact to predict Achievement, as opposed to constitute a single general variable—motivation. Interest showed strong, positive relationships between effort, utility, and self-efficacy. Self-efficacy, for its part, lead directly to students’ feelings of mathematical identity. However, the analysis revealed a perplexing finding regarding the role of interest and effort that is consistent with a number of previous studies, including Middleton et al. (in press). Namely, the combined effect of these two variables in predicting mathematics achievement only accounts for about 6% of the variation in achievement scores. It appears that motivation and achievement enjoy a more complex relationship than currently predicted by the literature.

The interrelationships among motivation variables and their impact on mathematics achievement in middle schoolers remains a complex issue. The current study shows demonstrably that key motivational variables interact significantly to influence students’ mathematics identity and achievement. In particular, the role of interest as a central construct connecting utility, effort and Selfefficacy, was supported. Other studies taking a structural equations modeling approach show similar findings. Singh, Granville, and Dika (2010), for example, show that effort, a composite attitude measure including items on usefulness and interest, and time spent on homework displayed direct effects on mathematics achievement.

However, in spite of 20 years of research by the author, promoting the importance of interest in the development of mathematics achievement, it appears that other variables have to play a critical role in addition to those currently hypothesized. Köller and Baumert (2001), for example, present a path model, examining the role of interest in predicting mathematics achievement in Grade 7, 10, and 12. They found that interest significantly predicted course selection, and through course selection, achievement was enhanced in each of the studied grades. Likewise, Marsh and Yeung (1998), show that mathematics self concept is better predicted by grades than test scores as indices of achievement. This indicates that grades are used as feedback for the development of academic self concept, but performance on external tests of achievement, since they are somewhat removed from the actual content of classrooms and grades, are more poorly predictive of motivational variables. The extent to which the HSLS:09 algebra assessment corresponded to students’ algebra experiences may have reduced potential sizes of effects of predictor variables.

Nevertheless, the consistently low path weights in the current study, Middleton et al. (in press), Schiefele et al. (1992), and Singh et al. (2010), show that something else must be influencing—mediating—interest, effort, and potentially other variables to influence middle schoolers’ achievement in mathematics.

What other candidates are there for players in this achievement game? Clearly this volume has evidence that classroom climate changes the feelings of belonging and the norms for productive mathematics learning (Jansen & Bartell, this volume; Megowan-Romanowicz et al., this volume). The recruitment of effective versus ineffective learning strategies by students, and the relationship of intrinsic and extrinsic orientations to this recruitment is also critical (McClintick-Gilbert et al., this volume). Affect, and intimacy with academic pursuits (Ely, Ainley, & Pierce, this volume; see also deBellis & Goldin, 2006) have been shown to be not just a byproduct of motivation and success, but as a key informational element in helping students choose what activities to expend effort in. It is also likely that these variables are critical to the development of self-efficacy and identity.

And we can also point to the role of the teacher in impacting middle schoolers’ mathematics motivation (Megowan-Romanowicz et al., this volume). Teachers who employ more learner-centered techniques may improve the social motivation in the classroom, making the learning more inclusive for more students. I must also add the quality of the curriculum and instructional resources for students, and presage variables such as family environment and socioeconomic status that tend to orient students toward academics in general and mathematics in particular. Put together, these variables represent a complex backdrop through which motivational variables must exert their influence.

Lastly, there is time. Middle schoolers’ mathematics beliefs have grown over a long period of time, orienting them to specific identities in relation to the content and the school’s treatment of that content. It may be that we must study the development of motivation over time, characterizing its change in relation to change in student learning and achievement as opposed to a single-time-point outcome such performance on an algebra test. Achievement, for example, affects interest—higher achievers tend to show more interest in mathematics, than low achievers, and this effect predicts future achievement. Future studies should examine the feedback between interest and achievement, perhaps mediated through effort, course choice, and other key student-related variables on the longitudinal growth of mathematical understanding.

The compendium of evidence shows that motivation and achievement are developmental, interdependent, and influenced by the design of educational experiences. In the middle grades, students tend to show less interest in mathematics, less mathematical self-efficacy, and poorer achievement over time (Gottfried, Marcoulides, Gottfried, Oliver, & Guerin, 2007). This problem is widespread in the United States, leading to a shortage of citizens willing to continue in STEM-related courses and careers. This suggests attention must be paid to implementing interventions at scale, designed to counteract this trend. The problem is, we do not yet know enough about how motivations develop over time, and their relation to the other critical variables that impact mathematics achievement. The advantage of utilizing the High School Longitudinal Study data for such research is that the study has collected information on demographics, economics, teacher behaviors, parent expectations and the like. Studied longitudinally across several panels, such information should shed light on the reasons students persist in mathematics and in STEM-related educational and occupational pursuits.