Relationships between elementary mathematics specialist certification, knowledge, beliefs, and classroom learning environments

Abstract

We report on the differences in mathematics learning environments in classes taught by certified Elementary Math Specialists (EMSs) (n = 28) and their peers (n = 33) as determined by observations of instruction. We used path analysis to examine how variables such as mathematical knowledge for teaching, beliefs, and background characteristics were related to the learning environment. We used the Classroom Learning Environment Measure (CLEM) observation protocol, which attends to aspects of mathematics lessons such as opportunities for students to justify their reasoning and attend to mathematical concepts. Our analysis revealed that learning environments incorporating such elements were significantly more prevalent in classes taught by EMSs, and that there were two paths indicating mediation effects on the relationship between EMS status and learning environment. One path was related to teachers’ beliefs about the primacy of computation in learning mathematics; the other path was related to teachers’ mathematical knowledge for teaching and their beliefs about the extent to which mathematical knowledge is constructed by the learner. We share implications for EMS programs and recommendations for future research on the impact of EMSs in elementary schools.

Introduction

Elementary Mathematics Specialists (EMSs) are professionals who have advanced preparation in mathematics teaching and learning at the elementary level (de Araujo et al. 2017; Lomas, 2022; Nickerson, 2010) and work in a variety of capacities to support elementary mathematics instruction (Baker et al., 2022a, 2022b; McGatha & Rigelman, 2017). In many cases, they have taken multiple years of graduate-level coursework (e.g., Harrington et al., 2017; Goodman et al., 2017), often in programs that are based on the 2013 Association of Mathematics Teacher Educators’ (AMTE) Standards for Elementary Mathematics Specialists or 2012 National Council of Mathematics Teachers’ (NCTM) Standards—Elementary Mathematics Specialist. These programs seek to develop strong knowledge of elementary mathematics, including the ways that student knowledge develops in particular domains (Sztajn et al., 2012), what representations can effectively portray and connect mathematical concepts (Ball et al., 2005), and how school policies and structures (including the use of textbooks, testing practices, teacher autonomy, leadership practices, etc.) can influence mathematics teaching and learning (de Araujo et al. 2017). Research suggests that EMS programs can have a substantial impact on teachers’ knowledge, as well as their beliefs about mathematics (e.g., Campbell & Malkus, 2014; Swars et al., 2018), and that EMSs can support improved learning outcomes for children (Kutaka et al., 2017; Rigelman & Lewis, 2023). A joint statement issued by AMTE, NCTM, the Association of State Supervisors of Mathematics (ASSM), and the National Council of Supervisors of Mathematics (NCSM) encouraged states, districts, and institutions of higher education to develop programs to prepare EMSs, advocating that “every elementary school have access to an EMS” (2022). Similar efforts are underway in the UK (Association of Mathematics Teacher Educators & National Centre for Excellence in the Teaching of Mathematics, 2014) and Australia (Australian Institute of Teaching & School Leadership, 2015; Driscoll, 2017; Lomas, 2022). In the USA, while many states have developed a process for obtaining formal EMS certification (Rigelman & Wray, 2017), in practice teachers are still often appointed to specialized roles for supporting mathematics instruction without specific preparation or credentials (Fennell, 2017).

Baker et al. (2022a, 2022b) found that most research on the impact of EMSs in schools has focused on their use as coaches or instructional leaders (e.g., Baker, 2022; Baker et al., 2022b; Bolyard & Baker, 2021; Campbell & Malkus, 2011; Campbell & Griffin, 2017; Harbour et al., 2018). But EMSs can serve a variety of roles, and even the coaching role can encompass a wide array of different activities (Campbell & Giffin, 2017; Driscoll, 2017). McGatha and Rigelman (2017) delineated additional roles for EMSs, such as serving as interventionists who work primarily with children or working as classroom teachers, in some cases assigned to teach mathematics to multiple classes of children throughout the day. Such “departmentalized” teaching assignments also have several varieties with different affordances and constraints (Markworth et al., 2016; Webel et al., 2017), and EMSs serving as classroom teachers can also engage in both formal and informal leadership activities (Conner et al. 2022), further blurring the lines between different roles and functions of EMSs.

When functioning as classroom teachers, EMSs have demonstrated increased capacity to advance the quality of learning opportunities for students (Myers et al., 2020, 2021; Nickerson, 2010) through, for example, the use of teaching practices described in NCTM’s recommendations (2000, 2014). These emphasize the use of multiple representations and solution strategies, the importance of students talking about mathematics with each other, supporting students’ use of conjectures, explanations, and justifications, and a sense of shared authority over the correctness of answers. All of these are related to the broader conception of mathematical learning as sensemaking (Hiebert et al., 1996) rather than answer-getting, and also to the broad findings that instruction that makes concepts explicit and engages students in some productive struggle is likely to lead to conceptual understanding (Hiebert & Grouws, 2007).

In general, the use of such practices is influenced by teachers’ beliefs and knowledge (e.g., Baumert et al., 2010; Campbell et al., 2014; Hill et al., 2008a, 2008b; Yurekli et al., 2020). Combined with findings that show higher levels of knowledge and more availing beliefs about teaching and learning for EMS teachers (Myers et al., 2020; Swars et al., 2018), it is reasonable to ask how these instructional practices are related to EMS certification, beliefs, and knowledge. Are certain kinds of beliefs sufficient for improved instruction, or are both beliefs and knowledge needed? Are there different ways in which having EMS certification is related to the development of certain types of mathematics learning environments (in particular, those types of environments supported by research and advocated for by professional organizations)? These are the questions we aimed to address in this study. Specifically, we first asked: “How do the classroom mathematics learning environments of EMSs differ from those of their peers?” Then, we asked, “What are the direct and indirect effects of beliefs, knowledge, and other background variables on the relationship between EMS certification and classroom learning environment?”.

Literature review and conceptual framework

There are many mathematical, pedagogical, and social aspects of a classroom learning environment that constitute children’s opportunities to learn mathematics (for a broad sampling, see Chapters 21–31 in Cai, 2017). Among these, there are varied interactions between the teacher, students, and the content (Cohen et al., 2003). However, the teacher has the primary responsibility for shaping the classroom environment by making instructional decisions about when and how members of the classroom will work and talk with each other (Walshaw & Anthony, 2008), what types of mathematical activities will serve as the focal points (Henningsen & Stein, 1997; Remillard, 2005), and how student work is used to move a lesson forward (Verschaffel et al., 2009). These practices are reflected in the specific actions of the teacher as well as in classroom norms that the teacher has either explicitly or implicitly fostered in the classroom. In the following sections, we detail some of the elements of the learning environment that have broad empirical support in terms of developing students’ conceptual understanding and positive dispositions toward mathematics, providing examples from studies of Elementary Mathematics Specialists if available. Then we discuss how researchers have used protocols to assess mathematics learning environments, including in classes taught by EMSs. Finally, we address the ways that knowledge and beliefs can influence how teachers establish learning environments and use this to describe the rough conceptual framework that we sought to refine in this study.

The mathematics learning environment

The mathematics learning environment is shaped in part by students’ engagement in specific types of mathematical activities like exploring conjectures and explaining and justifying solution strategies (Stylianides, 2007; Yackel & Cobb, 1996). When students’ justifications are brought to the attention of the whole class, teachers can use the opportunity to share mathematical authority with their students (Webel, 2010), a practice which elevates students’ mathematical reasoning and justification as a focal point for interactions. Students may be active in determining the validity of mathematical claims, deciding which mathematical questions are important, and guiding how such questions should be pursued by the classroom community (Ayala-Altamirano & Molina, 2021; Forman et al, 1998; Lannin, 2005).

In addition to providing support for exploring and validating conjectures, teachers can make pedagogical choices that position and encourage students to engage with each other’s ideas (Franke et al., 2015; Sherin, 2002; Woods et al., 2006). Based on student contributions, the teacher may adjust their instruction in the moment by opening up space for investigation of a student error (e.g., Schleppenbach et al., 2007), or by intentionally selecting students to share their work with the whole class based on mathematical features of their solutions or other sociomathematical reasons (Smith & Stein, 2011).

The learning environment is also characterized by teachers’ choices about how mathematical ideas and strategies are represented in the classroom. In particular, when multiple strategies or representations are presented, and explicit connections are made between these, students develop deeper understandings about mathematical relationships (e.g., Deliyianni et al., 2016; Rau & Matthews, 2017).

EMSs have demonstrated relatively high capacity to foster learning environments with the features described above (Myers et al., 2020; Myers et al., 2021; Rigelman & Lewis, 2023; Webel et al., 2018; Webel et al., 2022). For example, Myers reported EMS teachers’ improving in their capacity to facilitate productive mathematical discourse, including through the use of “effectively selecting and sequencing those students that share” and having student use “discourse to teach one another” (Myers et al., 2021, p. 320). In terms of using student thinking to inform instructions, Webel et al. (2018) described how an EMS used instructional videos with embedded questions to gather information about student thinking prior to class, which she then used to structure in-class discussions (e.g., setting up a debate regarding the different answers that were submitted). Another EMS, Keri, was observed pressing students to construct definitions and share multiple solution strategies, encouraging them to defend their conclusions with mathematical justifications (though, as we will note later, these practices were inconsistent across Keri’s multiple math classes) (Webel et al., 2022). This mixture of qualitative and quantitative research suggests that EMSs who complete formal training programs are relatively well-prepared to create the kinds of learning environments that have broad empirical support in terms of supporting students’ mathematics learning.

Assessing the learning environment

Several instruments have been developed to help researchers analyze particular aspects of mathematics classrooms (Boston et al., 2015), including those taught by EMSs. For this project, we adapted the Standards-based Learning Environment (SBLE)¹ protocol to develop what we refer to as the Classroom Learning Environment Measure (CLEM), explained in detail in the methods and provided in “Appendix A.” The SBLE has a track record of predicting student achievement in mathematics in the middle grades (Romberg & Shafer, 2003; Tarr et al., 2008) as well as in secondary schools (Grouws et al., 2013) and has been recently used in research with EMSs (Myers et al., 2020, 2021). While the tool has evolved over time, it’s consistent aim has been to assess the quality of learning opportunities that students have in mathematics classrooms, and it aligns well with the research on instructional practice cited above. For example, one indicator in the SBLE assesses the extent to which “the enacted lesson provided opportunities for students to make conjectures about mathematical ideas,” and another assesses the extent to which “students explained their responses or solutions strategies,” These indicators correspond to research findings about the importance of exploring conjectures and sharing justifications (e.g., Forman et al, 1998). Other indicators assess the extent to which “multiple perspectives/strategies were encouraged and valued” and “the teacher valued student’s statements about mathematics and used them to build discussion or work toward shared understanding for the class,” which aligns with research findings about the importance of sharing authority and using student thinking to shape instruction (e.g., Franke et al., 2015). The final indicator, “The enacted lesson fostered the development of conceptual understanding,” aligns with research indicating the importance of providing opportunities for students to make conceptual connections by, for example, comparing different strategies or representations (Deliyianni et al., 2016; Rau & Matthews, 2017).

Using the SBLE, Myers et al. (2020) found that there was variation in ratings of the learning environments established by EMSs on different indicators; for example, 92% of the teachers’ classes showed a high level of the indicator “the enacted lesson fostered the development of conceptual understanding,” while only 8% of the lessons showed a high level of the indicator, “the enacted lesson provided opportunities for students to make conjectures about mathematical ideas.” However, when combined into a composite score, 85% of EMS teachers demonstrated a generally high level of implementation of the practices assessed by the SBLE. Similarly, Tarr et al. (2008) found that SBLE ratings moderated the relationship between different types of curricula and student achievement; specifically, the use of Standards-based curricula was positively related to student achievement when there were moderate or high levels of Standards-based instruction as measured by the SBLE. This finding raises questions about how characteristics of teachers, such as their knowledge and beliefs, relate to the practices they use to establish learning environments. We turn to these next.

Teacher knowledge and instructional practices

Although specific formulations of mathematical knowledge for teachers (e.g., An et al., 2004; Ball et al., 2005; Marks, 1990) have been critiqued (e.g., Lannin et al., 2013; Charambalous et al., 2019; Copur-Gencturct, 2020), there is substantial evidence that mathematics teaching practice depends on a minimum level of knowledge (Hill et al., 2005, 2008a, 2008b), and that that knowledge includes both mathematical and pedagogical aspects (Hill et al., 2008a, 2008b). Teacher knowledge impacts practice because what teachers do when teaching mathematics depends on what they know how to do (Ball et al., 2005). Put differently, they cannot do things that require knowledge that they do not have, except perhaps unintentionally and sporadically.

In general, research has shown positive relationships between teacher knowledge and instructional practice (Baumert et al., 2010; Campbell et al., 2014; Hill et al., 2008a, 2008b; Kahan et al., 2003). However, this relationship is not always straightforward (Hill et al., 2008a, 2008b; Shechtman et al., 2010). Hill et al., (2008a, 2008b), for example, found that a teacher with low MKT taught more like a teacher with middle levels of MKT due to the influence of her textbook—she followed her textbook closely, leading to fewer mathematical errors than others with similar levels of MKT. Other teachers’ instruction was of lower quality than would be predicted from their MKT scores, and this the authors attributed to beliefs. Webel et al. (2022) described the case of an EMS who used teaching practices of different quality depending on which of her tracked classes she was teaching, though she demonstrated relatively high mathematical knowledge for teaching on a separate assessment. These cases suggest that while strong teacher knowledge is often necessary for ambitious mathematics teaching, it is not always sufficient.

Beliefs and instructional practice

A substantial body of literature indicates that teachers’ beliefs do inform their instructional practice (e.g., Speer, 2008; Stipek et al., 2001; Yurekli et al., 2020). Beliefs tend to be categorized in terms of beliefs about mathematics and beliefs about the teaching and learning of mathematics (Beswick, 2012; Philipp, 2007; Thompson, 1984). These beliefs are often interrelated. For example, Barkatsas and Malone (2005) found that teacher beliefs generally fell into two clusters: (1) a “contemporary—constructivist orientation,” which emphasizes human agency in the creation of mathematical objects and favors problem-driven instruction, a focus on mathematical practices/processes, and a collaborative approach to learning and teaching, and (2) a “mechanistic-transmission orientation,” which emphasizes of view of mathematics as static and objective, learning as receiving information, and teaching as transmitting information. The first cluster of beliefs are similarly generally associated with the kinds of mathematical learning experiences supported by research and advocated by the National Council of Teachers of Mathematics (2000, 2014) and cited above. The role of beliefs is underscored in that Principles to Actions (NCTM, 2014) lists beliefs that are consistent and in conflict with the eight practices endorsed in the document.

Another important type of belief is teachers’ attitudes about their ability to do and teach mathematics, sometimes described as self-efficacy beliefs (Enochs et al., 2000). Swars et al. (2018) found that participating in an EMS program led to shifts in beliefs, both in terms of teaching efficacy and a “cognitive orientation,” similar to the contemporary-constructivist orientation described above. Lomas (2022) reported similar findings for the Primary Mathematics Specialist Initiative in Victoria, Australia. This suggests that learning more about mathematics and how children learn mathematics can increase teachers’ confidence in their mathematics instruction.

While beliefs and practice often appear related, as with MKT, there are exceptions (Raymond, 1997; Webel et al., 2022; Yurekli et al., 2020), sometimes related to real or perceived institutional constraints. For example, Webel et al. (2022) described how the EMS Keri reported beliefs consistent with the “contemporary—constructivist” paradigm, yet she only employed practices associated with those beliefs in one of her tracked mathematics classes (the “highest” class). In her other classes, procedural performance was prioritized over discussion of multiple strategies and representations, driven by Keri’s perceptions regarding the needs of students in those classes and her obligations to them. This suggests that even with productive beliefs about mathematics, teaching, and learning, teachers may choose to use practices that lead to learning environments that do not align with those beliefs.

A conceptual framework for the current study

As noted, Elementary Mathematics Specialist Programs are designed to influence beliefs, knowledge, and instructional practice (de Araujo et al. 2017). Our conceptual framework is grounded in the hypothesis that EMSs, by virtue of their decision to pursue EMS certification as well as the effect of their learning experiences in their programs, will teach in ways that are more consistent with the practices described in the literature reviewed above. For example, they will be more likely than their peers to establish learning environments in which students’ ideas drive discussions, multiple strategies and representations are shared and connected, and justifications are provided for mathematical conclusions (Myers et al., 2020, 2021). We also hypothesize, based on the research described above, that mathematical knowledge for teaching and beliefs and attitudes may mediate these relationships, but the research also suggests that these relationships are not straightforward. Figure 1 represents an “unrefined model,” presenting an opportunity to clarify how knowledge/belief/attitude variables might interact with each other and how they might mediate the relationship between EMS certification and learning environment.

Fig. 1

Conceptual framework (unrefined)

Other projects have explored similar questions using similar frameworks (e.g., Ernest, 1989; Wilkins, 2008; Yang et al., 2020); for example, Wilkins (2008) found that content knowledge, attitudes, and beliefs, were all related to inquiry-based instructional practice (self-reported), and that beliefs mediated the relationship between content knowledge and instructional practices. However, we have not located a study focused on EMSs that quantitatively examines relationships between knowledge, beliefs (including self-efficacy), and instructional practice. The closest such investigation is Myers et al. (2020), which found that teacher knowledge increased over the course of an EMS program and that participants were employing teaching practices consistent with the principles advocated for in the program. They used interview data to generate profiles of shifts individual teachers made over the course of the program. However, there was no quantitative analysis of the relationships between these variables, and there was no comparison group. Kutaka et al. (2017) did analyze growth in multiple domains of EMS participants’ knowledge, confidence, and student-centered beliefs about the teaching and learning of mathematics, and used a comparison group. However, that study did not measure instructional practice. In the present study, we used observations to document the classroom learning environments (including instructional practices) in mathematics lessons taught by EMSs and used path analysis to clarify the ways that EMS certification (e.g., via knowledge and beliefs) was related to these learning environments.

As we have noted, the relationships in our framework are complex, and beliefs and knowledge do not always relate to practice in the way one might expect. We do not know, for example, whether certain beliefs will be more strongly related to learning environment, or whether MKT will be more strongly related to learning environment than beliefs. By analyzing the role of beliefs, knowledge, practices, within the context of a study including both EMSs and their peers, we hope to better understand the interplay of these variables. In particular, we are interested to first confirm whether EMSs in our sample established different kinds of learning environments than their peers, and then examine how their knowledge and beliefs may have mediated these differences.

Methods

Context

Participants were recruited from a pool of graduates of a two-year EMS program that was co-designed across five institutions in the state of Missouri and included 24 credits of graduate-level coursework aligned with the AMTE Standards for Elementary Mathematics Specialists (2013). The coursework led to EMS certification granted by the Missouri Department of Elementary and Secondary Education. After developing the program, the coalition of faculty continued to meet biannually to revise courses and discuss programmatic issues (recruitment, communications with state education administrators, etc.) for the next several years (Goodman et al., 2017). The courses themselves were blended, with online coursework combined and five face-to-face sessions each semester (20 total over the course of the program). There were five content courses, each focused on developing deep knowledge of elementary mathematics concepts, awareness of how children develop this knowledge, and engagement with the kinds of tasks, representations, and discourses that support mathematics learning. There were also two leadership courses, which addressed the history of mathematics education, the role of textbooks and curricular programs, general leadership, and specific mathematics leadership skills like coaching teachers, facilitating professional development, interpreting standardized testing data, co-teaching, conducting lesson studies, and negotiating duties with school administrators.

Participants

Participants included 3rd, 4th, or 5th grade teachers (n = 28) who had completed an EMS program at one of the five institutions in the state of Missouri. Each year of the study, recruitment began by sending each graduate of the EMS program a survey asking for information regarding their teaching assignment. Based on responses, all teachers who would be teaching in grades 3 through 5 with at least some responsibility for mathematics in the coming year were invited to participate via a letter with follow-up phone calls. This process led to the identification of 28 teachers who satisfied the eligibility criteria (completed EMS program with teaching assignment including mathematics at grade 3, 4 or 5) and agreed to participate in the study.

A group of comparison teachers (not EMS certified) (n = 33) was also recruited for the study. These teachers were recruited from the same (or nearby) schools and grade levels as the EMS participants. In order to increase the power of the analysis and to address potential attrition, we recruited multiple comparison teachers, if available, for each EMS, resulting in a larger number of non-EMS participants. Table 1 shows the participating teachers’ demographics, prior professional experience, and additional characterizations of their professional background. Note that several teachers participated in the project for two years; this manuscript only reports each teacher’s first year data.

Table 1 Professional background and experience of participating teachers

Study design

Because teachers could not be randomly assigned to the EMS condition, this research utilized a quasi-experimental design. That is, teachers were selected by EMS certification, and we statistically controlled for the background variables listed in Table 1. Similar to other projects studying EMSs as classroom teachers (Kutaka et al., 2017; Myers et al., 2020; Swars et al., 2018), we collected data on knowledge and beliefs of participants. Like Myers et al. (2020), we used classroom observation ratings to characterize the learning environment, and like Kutaka et al., (2017), we had a group of non-EMS teachers to serve as a comparison.

To answer the research questions, we performed several different quantitative analyses. For RQ1 (how do the classroom mathematics learning environments of EMSs differ from those of their peers?), we first compared EMS and non-EMS scores on individual elements of the Classroom Learning Environment Measure (CLEM) observation protocol and an overall observation factor score (described below). For RQ2 (what are the direct and indirect effects of beliefs, knowledge, and other background variables on the relationship between EMS certification and classroom learning environment?), we conducted a series of multiple regression analyses (path analysis) to examine the relationships between observation scores and teacher variables. This approach is similar to previous studies that investigated the relationships between teacher knowledge, beliefs, and practices (e.g., Wilkins, 2008; Yang et al., 2020).

Measures

3.4.1. Classroom observations

As previously mentioned, the CLEM observation tool was adapted from the SBLE tool (e.g., Grouws et al., 2013; Romberg & Shafer, 2003; Tarr et al., 2008). Specifically, in order to capture a greater variety in instructional practice, we increased the scale from three points (1–3) to five points (0–4), with detailed rubrics for the anchor levels of 0, 2, and 4. We also adapted the descriptions to more specifically relate to elementary level content (e.g., written examples of each anchor level were constructed based on clips from a national repository of videos of elementary mathematics instruction—see “Appendix B” for an example for the indicator, “Student thinking was used to adjust instruction”). Finally, we expanded two indicators (“Multiple perspectives/strategies were encouraged and valued” and “Students explained their responses or solution strategies”) into five indicators in order to better reflect the research on learning environments and instructional practices. That is, we sought to capture with more specificity the kinds of ideas that might be shared in a class conversation, such as strategies (element SM1 in “Appendix A”), justifications (R2), and representations (SM3), as well as the extent to which student thinking was used to adjust instruction (ST2) and how classroom authority was distributed across the learning environment (R3). The final protocol consisted of field notes on the lesson, scores for each of the eight elements of the learning environment, and brief rationales for each score with evidence linked to the field notes.

Initial training for the project team on the observation instrument was conducted in a three-day workshop during the summer prior to the first year of data collection. Using mathematics lessons in the ETS Classroom Video Library (http://etsvideos.mylearningplan.com/support/home), the project team selected clips and full lessons that embodied a range of quality in the eight elements of the CLEM. During the training workshop, we used a series of video clips that embodied the full range of anchor scores (0, 2, 4) for each of the eight elements, and these stimulated rich discussions on coding decisions. On the final day of training, observers viewed two full-length mathematics lessons that ran uninterrupted, individually applied the scoring rubric to assign codes for each of the eight elements and discussed their codes in a whole-group setting facilitated by co-developers. Feedback from the training workshop led to modest changes in the coding descriptors (e.g., replacing “some” with “at least one such instance”) to enhance clarity and promote the calibration of codes across observers.

Observation data were subsequently collected over a period of three years, with three observations for each participating teacher in each year. In the first year, all lessons were observed live and coded independently by two trained observers, who later met to resolve discrepant scores. The original scores assigned by each observer were analyzed, and 84.8% of codes differed by no more than 1 (on a 5-point scale), providing evidence of relatively high inter-rater reliability. In Years 2 and 3, approximately one-third of all lessons were randomly selected for double-coding to gauge consistency. Across all three years of data collection, 87.0% of codes assigned in these lessons differed by no more than 1 point, indicating substantial agreement between observers. There were no video recordings made of the observed lessons.

To identify the number of potential latent traits under the CLEM, we conducted an exploratory factor analysis (EFA) using the ‘psych’ package in the R software (Revelle, 2020). This analysis was performed using data from teachers’ first year of participation in the study (n = 61). Based on parallel analysis and analysis of the scree plot, a one-factor model was identified as the most appropriate for the CLEM elements. For element SM3 (“Connections between multiple types of representation were made”), the communality was below 0.20 for two of the three rounds, and so this element was excluded from further analysis.² With the remaining seven elements, this single factor model demonstrated strong internal consistency (α = 0.95) (Table 2). A single observation score based on the remaining seven elements was created for each teacher by averaging the scores for each element across all three rounds of observation and summing these average scores.

Table 2 Factor loadings for each of the CLEM indicators

Knowledge for teaching

During the summer prior to the beginning of the school year, we administered four subtests of the Learning Mathematics for Teaching (LMT) instrument, a widely used tool for measuring mathematics knowledge for teaching and with demonstrated sound psychometric properties (see Schilling et al, 2007). The subtests were Number and Operations Grades K-6; Patterns, Functions, and Algebra Grades K-6; Geometry Grades 4–8; and Data, Probability, and Statistics Grades 4–8. Each subtest consists of 20 items focused specifically on mathematical knowledge for teaching (e.g., the validity of solution strategies and representations, using mathematical definitions, representing mathematical content to students, and identifying adequate mathematical explanations). In this project, for each teacher we computed an average score across the four subtests to use in subsequent analyses.

3.4.4. Beliefs

We administered a survey developed by White et al., (2005/2006) to measure elementary teachers’ beliefs and attitudes. The beliefs and attitude scale consisted of 38 items with a five-point Likert scale:1 = strongly disagree to 5 = strongly agree (“Appendix C”). To consolidate the 38 items about beliefs and attitudes from White et al., (2005/2006), we used EFA³ and identified four latent factors based on a scree plot and the oblique rotation method. The first factor (Constructing) included items related to the belief that mathematical knowledge is constructed; the second factor (Computation) included items related to the belief that knowing mathematics is mostly about knowing how to compute the right answer quickly; the third factor (Security) consisted of items about the attitude of security in teaching mathematics, and the fourth factor (Confidence) reflected the attitude of confidence in doing mathematics. Table 3 lists sample items for each factor, and the full list of items is included in “Appendix C.” The inter-factor correlations demonstrated a low correlation between factors (“Appendix D”). Based on the EFA result, we generated a weighted factor score for each factor as a teacher-level variable for further analysis.

Table 3 Factor loadings and sample items for beliefs and attitudes survey

Data analysis

First, we compared the mean scores of certified EMSs and the comparison group on the subtests of the LMT and their responses to the beliefs survey using Welch’s two-sample t-test, checking normality (Shapiro–Wilk test) and homogeneity of variance (F-test). For RQ1, we then compared the two groups with regard to the individual indicator scores on the CLEM observation protocol as well as the overall observation factor score. For RQ2, we performed a preliminary regression analysis to investigate the relationships between observation scores and EMS certification, teaching assignment, beliefs, attitudes, and knowledge as well as several teacher and school background variables. Based on the regression results, we performed path analysis using the ‘lavaan’ package in the R software (Rosseel, 2012) to confirm the overall structural relations in the model. The model was evaluated by the Tucker-Lewis index (TLI), Comparative Fit Index (CFI), and root mean square error (RMSEA). The selection of the final model was based on the recommended cut-off values by Hu and Bentler (1999): TLI ≥ 0.95, CFI ≥ 0.95, and RMSEA ≤ 0.05.

Results

Descriptive data for beliefs and MKT are shown in Table 4, with significant differences as determined by t-tests indicated. Since six variables (Number & Operation, Algebra, Geometry, Probability, Computing, Confidence) satisfied assumptions of normality and homogeneity of variance, we adopted Welch’s two-sample t-test. Since Constructing and Security did not satisfy assumptions of normality, we adopted the Wilcoxon rank sum test with two variables as an alternative to Welch’s t-test. The EMS teachers had higher scores than their non-EMS peers on all four of the LMT subtests. There were statistically significant differences for reported beliefs about mathematical knowledge as constructed by learners (Constructing factor), as well as the Computation factor, which measured the belief that knowing mathematics means being able to compute or obtain the answer quickly. For example, EMSs averaged 2.50 (on a scale from 1 to 5) for the statement, “being able to memorize facts is critical in mathematics learning,” compared to 3.65 for non-EMS. EMSs also indicated higher levels of security about teaching mathematics (Security) and confidence in doing mathematics (Confidence).

Table 4 Descriptive data for MKT and beliefs

Learning environment

Our analysis revealed significant differences between EMSs and their peers across the seven retained indicators on the observation protocol, with effect sizes ranging from 0.628 to 0.893 (Table 5). Factor scores on the observation instrument revealed that, on a scale from 0 to 28, teachers with EMS certification earned higher scores than their peers by over 5 points on average. EMSs’ scores on all elements except for R1 fell in the upper half of CLEM observation rubrics (i.e., scores above 2), while non-EMS teachers scored below the midpoint of the observation rubrics for all elements except SM2, for which their average score was marginally above the middle score.

Table 5 Means (Standard Deviations) for EMS and non-EMS and effect sizes

Relationships between the learning environment and EMS certification, beliefs, knowledge, and other background variables

We ran four regression models (Table 6) to examine the relationships between observation scores and the following variables: EMS certification (model 1), background variables (years of teaching experience, advanced degree, grade level, school size, school FRL percentage, and school title 1 status) (model 2), MKT (model 3), and beliefs/attitudes (model 4). The variables that significantly predicted observation scores were EMS certification, MKT, and two beliefs scales (Constructing and Computation). None of the background variables in Model 2 predicted observation scores, nor did either of the two attitude scales in Model 4.

Table 6 Results from regression analysis

To examine the interrelationships between these five variables (i.e., EMS, MKT, Constructing belief, Computation belief, and Observation scores), we conducted a path analysis. Based on our theoretical framework, we positioned EMS certification at the beginning of the path, and learning environment at the end of the path, and then tested paths through the remaining three variables (MKT, Constructing belief, and Computation belief) (see Table 7).

Table 7 Path analysis results

As shown in Table 7, the initial model (Path 1), with all variables included and observation scores as the outcome, showed that Computation and Constructing significantly predicted observation scores. Subsequent tests revealed three other significant results. Path Model 2 shows that MKT predicts the Constructing belief. Path Model 3 shows that EMS certification predicts the Computation belief, and Path Model 4 shows that EMS certification predicts MKT. These relationships were combined to construct the full path model shown in Fig. 2, which demonstrated an acceptable model fit (chi-squared (df) = 5.188 (4), p-value = 0.269, CFI = 0.986, TLI = 0.964, RMSEA = 0.07). Indirect effect sizes for each relationship are shown in Table 8.

Fig. 2

Parsimonious Path Model. Standardized coefficients (β) were reported. All paths were significant at the alpha level = .05

Table 8 Mediation Effect with 95% confidence interval

In summary, the path analysis determined there were two primary paths through which EMS certification was related to the learning environment. The first path was through MKT (as measured by the LMT) and then through the Constructing belief. That is, teachers with EMS certification exhibited higher levels of mathematical knowledge for teaching, teachers with higher MKT showed higher levels of the Constructing belief, and teachers with higher scores on those belief items had generally higher CLEM scores. These findings are in line with our hypotheses about the role of both knowledge and beliefs in establishing the learning environment.

The second path did not go through MKT. It showed that another way that EMS certification was related to learning environment was through the Computation belief—the belief that doing mathematics is primarily about computational skills. Teachers with EMS certification less often agreed with these statements (i.e., they scored lower on this belief scale), and teachers with low scores on this belief had significantly higher classroom environment ratings. This finding also aligns with our hypotheses about the influence of beliefs on instruction, though we find it interesting that MKT was not a significant variable in this second path.

Summary

In summary, the path analysis showed two distinct ways that EMS certification was related to the presence of elements of the learning environment measured by the CLEM. The first path showed that EMS status was related to higher LMT scores, which was related to beliefs that students construct their own mathematical knowledge, which was related to higher CLEM ratings. The second path showed that EMS status was negatively related to the belief that mathematics is primarily about computation, which was negatively related to higher CLEM ratings. These paths help to refine our understanding of how EMS certification relates to instructional practice, and how specific types of beliefs and knowledge mediate this relationship.

Discussion

In this study, we found that EMS certification was consistently and strongly related to differences in the classroom learning environment, reflected by instructional practices such as adjusting instruction based on student thinking, sharing mathematical authority, and pressing students to justify their reasoning. There were a few caveats in our observation data, however. For example, across all classes there was generally a dearth of opportunities for students to explore conjectures, a finding similar to that of Myers et al. (2020). We surmise that exploring conjectures is atypical in elementary mathematics teaching and it does not figure prominently in teachers’ vision or planning for instruction, even for EMS teachers (although there was still a significant difference on that criterion between EMSs and their peers). Also, the use of multiple representations was the only indicator that did not show a significant difference between EMSs and their peers. One possibility is that representation use is significantly dependent upon the mathematical content of a given lesson, which was not controlled for in our observation schedule.

Furthermore, we found that there were two distinct paths by which EMS certification was related to learning environment. Specifically, the first path showed that EMS certification was related to mathematical knowledge for teaching and the belief that mathematical knowledge is constructed by students, which was related to learning environment. The second path showed that EMS certification was related (negatively) to the belief that mathematics is primarily about computation, which is related (negatively) to learning environment.

Contributions

These findings add to our understanding about how expertise is related to instructional practice. While we cannot claim that the EMS program in our program caused the EMS teachers to have higher levels of MKT or different beliefs from their peers, we can say that EMS teachers were different in characteristics that were predictive of differences in our observation scores. Other research does suggest that, in fact, EMS programs can have a positive influence on teachers’ beliefs and practice (Kutaka et al., 2017; Myers et al., 2020; Swars et al., 2018).

In a sense, these findings align with what we might expect, regardless of whether the differences were caused by the EMS training or were a result of teachers with higher levels of MKT or more productive beliefs about mathematics self-selecting into EMS programs. But the two paths give us additional insights into what elements of expertise are connected to differences in practice. We were surprised, for instance, that there was a path where the computation belief was the sole mediator. This suggests that meaningful differences in teaching practice were connected to participating teachers’ core conceptions about the purpose of learning mathematics, independently of their knowledge of student mathematics or other beliefs. We are struck by the explanatory power of the simple, persistent idea that the purpose of teaching mathematics is to enable students to accurately perform calculations (Thompson et al., 1994).

We are less surprised by the other path yielded by our analyses. EMS programs are designed around the idea that teachers’ understanding of student mathematics, and their belief in students’ ability to construct their own ideas, has a large influence on their teaching practices (de Araujo et al. 2017). Accordingly, it makes sense that EMS teachers are more likely to have this knowledge and these beliefs, and that teachers with this kind of expertise would engage in practices like adjusting their instruction based on student thinking, using multiple strategies and representations, and sharing mathematical authority with students. This is consistent with other research showing relationships between knowledge and practice (e.g., Copur-Gencturk, 2015; Hill et al., 2008a, 2008b), but adds a connection to teachers with EMS certification.

Implications

Our work complements other research that shows that changes in teaching can occur with sustained attention to the mathematics that is being taught and what teachers believe about that mathematics and what it means to learn and teach that mathematics (e.g., Fennema et al, 1996; Myers et al., 2020). One implication of this project is the seemingly prominent role played by the belief that mathematics learning is primarily about answer-getting (Hiebert et al., 1996). This is not simply a cultural value among teachers of mathematics; it is pervasive among students (Muis, 2004) and presumably, given their educational experiences, many adults. Although we cannot say from this study what is required to alter those beliefs, we can say that teachers who were less concerned about answer-getting were more often engaged in establishing the research-based learning environment measured by our observation protocol.

Second, our work showed that the EMS teachers in our project had higher levels of mathematical knowledge for teaching and stronger beliefs that mathematical knowledge is constructed than their peers, and those teachers were more often observed pressing students for mathematical justifications and incorporating student thinking into their classroom discourse, practices which we have documented qualitatively among our participants elsewhere (Webel et al. 20182022). This finding complements other research showing relationships between mathematics-focused programs for in-service elementary teachers, knowledge and beliefs, and practice (Hill & Ball, 2004; Kutaka et al., 2017; Myers et al., 2020; Swars et al., 2018).

In more practical terms, we suggest that if school administrators want teachers to engage in the instructional practices advocated by NCTM (2014), they should look for teachers with the kinds of knowledge and beliefs that were identified in our path analysis, or look for ways to develop this knowledge and these beliefs in their current teachers. In our sample, EMSs exhibited these characteristics more regularly than their peers. Combined with research showing the potential of using EMSs as instructional leaders and coaches (Baker, 2022; Baker et al., 2022b; Bolyard & Baker, 2021; Campbell & Malkus, 2011; Campbell & Griffin, 2017; Harbour et al., 2018), our findings support the use of EMSs in elementary school settings, as either classroom teachers or leaders. Because EMSs more often display the knowledge and beliefs to enact effective instruction, they are more likely to have the capacity to support their peers in enacting those practices (Rigelman & Lewis, 2023), though it is important to recognize that coaching requires different expertise than teaching (Chval et al., 2010). Indeed, some of the EMSs in our study were able to exhibit both formal and informal leadership in mathematics without leaving their primary teaching responsibilities, though some activities were more likely to support teacher learning than others (Conner et al. 2022).

This study also raises questions for future research. While different research projects have shown individual relationships between different characteristics of EMSs, their practice, and their impact, we lack a comprehensive study that connects EMS programs, beliefs, knowledge, learning environment, and student achievement. This would require a research design that controls for differences between teachers prior to their enrollment in an EMS program, perhaps using random assignment or a waitlist design in order to establish causality. We suggest that the preliminary evidence is sufficiently strong to warrant the substantial investment needed to undertake such a large-scale investigation. In addition, as EMS teachers become more prevalent, researchers may consider including EMS certification as a variable in studies of elementary mathematics teaching.

Conclusion

This study addressed the relationship between EMS certification and learning environments that exhibit the kinds of characteristics recommended by the National Council of Teachers of Mathematics (2014), examining mathematical knowledge for teaching and beliefs as mediating variables. We found that EMS teachers more regularly used NCTM-recommended practices, had higher levels of MKT, and beliefs less focused on computational ability as a primary goal for mathematics instruction. We found that there were two paths by which EMS certification related to learning environments, one via beliefs about computation and one via MKT and beliefs about mathematical knowledge as constructed. The study was limited by a relatively small number of participants, although this is not uncommon for research on Elementary Mathematics Specialists (e.g., Campbell & Malkus, 2011; Myers et al., 2020). Although data collection spanned multiple years, our analysis was limited to a single year of data for each participant; Campbell and Malkus (2011) found a delayed effect of coaches on student achievement, and future research could take a more longitudinal approach and include qualitative data such as narratives or interviews. Nevertheless, this research helps shed light on a complex constellation of variables related to instruction and provides some suggested priorities for addressing instructional quality. It also points to the need for comprehensive study of EMS programs that include attention to beliefs, knowledge, learning environment, and effects on student outcomes.

Appendices

Appendix A: The classroom learning environment measure (CLEM) protocol

Element	0	2	4
R1. Students were afforded opportunities to formulate and investigate conjectures about mathematical ideas	Students had few, if any, opportunities to investigate conjectures in the lesson	Students had opportunities to investigate conjectures offered by the teacher	Students had opportunities to formulate their own conjectures and investigate the validity of those conjectures
R2. Students created and defended mathematical justifications	Students were afforded few, if any, opportunities to create or share mathematical justifications	Students’ mathematical justifications were seldom challenged by the teacher or other students, or this generally occurred only when faulty reasoning was offered	Students’ mathematical justifications were challenged by the teacher or other students. Students responded to questions or critiques of their reasoning
R3. Mathematical authority was shared by members of the classroom community	Students relied on the teacher as the primary authority and as the source of mathematical knowledge. The teacher solely determined the validity of mathematical contributions or prompted students to refer to the textbook	Students were encouraged to consider the validity of at least some statements but in a superficial way. The teacher at least initially withheld judgments about the mathematical validity of students’ reasoning or answers but ultimately asserted authority	Students were responsible for discussing the validity of at least some statements. The teacher withheld judgments about the mathematical validity of students’ reasoning or answers and instead prompted student involvement
ST1. Evidence of student learning was used to adjust instruction	The teacher sporadically or superficially elicited evidence of student learning by posing questions, making observations, and listening to students’ thinking. The teacher generally accepted student responses and moved on. Instruction did not appear to be directly adjusted based on student work, questions, and responses	The teacher elicited evidence of student learning by posing questions, making observations and listening to students’ thinking, and generally used student responses to continue discussion. Although instruction generally did not appear to be adjusted based on student work, questions, and responses, there may have been at least one such instance	The teacher actively elicited evidence of student learning by posing questions, making observations, and listening to students’ thinking. The teacher purposefully selected students to share their thinking and instruction appears to be adjusted in the moment based on student work, questions, and responses
ST2. Students' statements about mathematics were used to advance discussions	The teacher poses questions generally with a specific response in mind. Students respond by stating facts, definitions, or procedures. Students' responses were not typically used to advance discussion. Connections between students’ statements, or between students’ statements and mathematical ideas were generally not made	The teacher directs classroom discussion by posing questions intended to lead students down a particular path of discussion. The teacher uses desired responses to advance classroom discussion, and either ignores or directly addresses other responses. Connections between students’ statements, or between students’ statements and mathematical ideas are sometimes made	The teacher facilitates classroom discussion by pressing students to communicate their thoughts clearly, and expecting them to reflect on their thoughts and those of their classmates. The teacher uses particular contributions to advance discussion. Connections between students’ statements, or between students’ statements and mathematical ideas are evident
SM1. Multiple (alternative) solution strategies were discussed	Different perspectives or strategies for solving problems did not surface or were not valued. If students volunteered alternate approaches, the teacher responded to the student directly and moved on. Generally, if a student offered a correct solution, the teacher accepted it and moved on	Different perspectives or solution strategies occasionally surfaced but primarily occurred when another student had not yet mentioned a particular solution method. Multiple strategies are primarily seen as disjoint options for solving a problem, and class discussion focused on using prescribed approaches	Students viewed problems from multiple perspectives. When appropriate, alternative entry points or solution strategies were solicited and discussed. Connections between the varying approaches were made explicit in-class discussions
SM2. The enacted lesson developed mathematical knowledge in meaningful ways	The focus of mathematical knowledge was on algorithms and procedures, formulas and definitions without meaning. Typically, information was presented to students without discussion of mathematical connections, development of concepts, or components	The focus of mathematical knowledge was on algorithms and procedures, formulas and definitions with some attention to meaning. Information was presented with some discussion of mathematical connections, development of concepts, or components. Verification of new ideas tended to focus on how (but not why) the mathematics “works.”	The focus of mathematical knowledge was on algorithms and procedures, formulas and definitions with strong attention to meaning. Mathematical concepts were developed through the generalization of existing concepts with a primary focus on understanding their components, relationships among them, and why the mathematics “works.”
SM3. Connections between multiple types of representation were made	The lesson generally did not emphasize multiple types of representation of mathematical concepts and procedures. The teacher primarily focused on singular (typically symbolic) representations of ideas and did not elicit, use, or make connections to other representational forms	The lesson elicited multiple types of representation of mathematical concepts and procedures. Although different representational forms occasionally surfaced, there was little discussion about explicit connections among representations	The lesson emphasized using and making connections among types of mathematical representation to deepen student understanding, support classroom discourse, and serve as tools for solving problems. When appropriate, students used, discussed, and made connections among contextual, visual, verbal, physical, and/or symbolic representational forms

Appendix B: An example of a set of written vignettes that accompanied the observation scoring rubrics

ST.1 Evidence of student learning was used to adjust instruction.

0	1		2	3		4
The teacher sporadically or superficially elicited evidence of student learning by posing questions, making observations, and listening to students’ thinking. The teacher generally accepted student responses and moved on. Instruction did not appear to be directly adjusted based on student work, questions, and responses		The teacher elicited evidence of student learning by posing questions, making observations and listening to students’ thinking, and generally used student responses to continue discussion. Although instruction generally did not appear to be adjusted based on student work, questions, and responses, there may have been at least one such instance			The teacher actively elicited evidence of student learning by posing questions, making observations, and listening to students’ thinking. The teacher purposefully selected students to share their thinking and instruction appears to be adjusted in the moment based on student work, questions, and responses
Look-For: The teacher presents the problem: 960 people need to use shuttle busses. There are 8 busses which hold 40 people each. How many trips must each bus make before everyone reaches their destination? After giving students a few minutes to work on the problem, the teacher presents them with 4 possible solutions. For each solution, the teacher asks how many students arrived at that answer. The teacher identifies that 3 is the correct answer, demonstrates multiple solution methods, and begins the next activity		Look-For: The teacher presents the problem: 960 people need to use shuttle busses. There are 8 busses which hold 40 people each. How many trips must each bus make before everyone reaches their destination? After giving students 5 min to work on the problem, the teacher selects a student who has the correct answer to show her work on the board (they found 960 ÷ 8 and divided by 40). The teacher asks, “Did anyone get the same answer a different way?” One student who did is called up to the board to share his answer (he found 960 ÷ 40 and divided by 8). No students got the correct answer using a different method, so the teacher shows students a third method (divide 960 by the product 40 × 8)			Look-For: The teacher presents the problem: 960 people need to use shuttle busses. There are 8 busses which hold 40 people each. How many trips must each bus make before everyone reaches their destination? After giving students a few minutes to work on the problem, the teacher selects 3 students who found the correct answer of 3 in different ways to share their work. (e.g., “I noticed Jamie did something a little different. Do you mind sharing what you did with us?”) One student is unconvinced by these explanations and suggests the answer is 20. The teacher has this student share his work (they added 40 and 8, then divided 960 by 48), then leads the class in a discussion to clarify why the proposed method is incorrect
Rationale: In the case of 4, the teacher observes students as they solve the problem, and selects students who used different methods. Rather than directly correcting a student error, the teacher adjusts the lesson by having him share his thinking and prompting the class discuss this approach. Similar adjustments were made at other times during the lesson. For the lesson coded 2, the teacher allows students to share their thinking, withholding alternate solution methods until the students finished. Although students share solutions throughout the lesson, errors are not typically examined and the teacher does not appear to deviate from the plan. In the case of 0, the teacher seeks superficial evidence of student learning by checking answers. Although this may occur in any lesson, there are few instances of instructional segments in this lesson that deviate from this pattern

Appendix C: Beliefs and attitudes scales (items from White et al, 2005/2006)

Belief: Mathematical knowledge is constructed (α = 0.86).

Items	Loading
12: Mathematics learning is enhanced by activities which build upon and respect students’ experiences	0.771
4: Mathematics is a beautiful, creative and useful human endeavor that is both a way of knowing and a way of thinking	0.705
14: Teachers should provide instructional activities which result in problematic situations for learners	0.682
9: Periods of uncertainty, conflict, confusion, surprise are a significant part of the mathematics learning process	0.645
18: Teachers should negotiate social norms with the students in order to develop a co-operative learning environment in which students can construct their knowledge	0.607
3: Mathematics is the dynamic searching for order and pattern in the learner’s environment	0.598
17: Teachers should recognize that what seem like errors and confusions from an adult point of view are students’ expressions of their current understanding	0.571
6: Mathematics knowledge is the result of the learner interpreting and organizing the information gained from experiences	0.534
10: Young students are capable of much higher levels of mathematical thought than has been suggested traditionally	0.424
13: Mathematics learning is enhanced by challenge within a supportive environment	0.403

Belief: Mathematics is computation (α = 0.76).

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2: Mathematics problems given to students should be quickly solvable in a few steps	0.654
15: Teachers or the textbook—not the student—are the authorities for what is right or wrong	0.634
1: Mathematics is computation	0.607
11: Being able to memorize facts is critical in mathematics learning	0.531
8: Mathematics learning is being able to get the right answers quickly	0.426
5: Right answers are much more important in mathematics than the ways in which you get them	0.411
16: The role of the mathematics teacher is to transmit mathematical knowledge and to verify that learners have received this knowledge	0.338

Attitude: Security in teaching mathematics (α = 0.94).

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19: Of all the subjects, mathematics is the one I worry about most in teaching	0.913
6: I do not enjoy having to teach mathematics	0.789
4: I’m not the type of person who could teach mathematics very well	0.780
17: It wouldn’t bother me to teach a lot of mathematics at school	0.775
13: Teaching mathematics doesn’t scare me at all	0.731
20: If I taught in a team or with a teaching partner, I’d like to have another teacher teaching the mathematics	0.728
12: I would get a sinking feeling if I came across a hard problem while teaching mathematics	0.641
3: Mathematics makes me feel inadequate	0.635
1: Generally I feel insecure about the idea of teaching mathematics	0.605
10: Time passes quickly when I’m teaching mathematics	0.602
2: I find many mathematical problems interesting and challenging	0.544
15: I am confident about the methods of teaching mathematics	0.502
16: I have trouble understanding anything that is based upon mathematics	0.497
9: I’m not sure about what to do when I’m teaching mathematics	0.480

Attitude: Confidence in doing mathematics (α = 0.88).

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5: I have always done well in mathematics classes	0.820
8: I have generally done better in mathematics courses than other courses	0.790
7: I am quite good at mathematics	0.722
14: At school, my friends always came to me for help in mathematics	0.633
18: I never do well on tests that require mathematical reasoning	0.566
11: I have hesitated to take courses that involve mathematics	0.505

Appendix D: Inter-factor correlation from the exploratory factor analysis

	Constructing	Computation	Security	Confidence
Constructing	1	0.23	0.41	− 0.23
Computation	0.23	1	0.08	− 0.28
Security	0.41	0.08	1	− 0.07
Confidence	− 0.23	− 0.28	− 0.07	1