Abstract
Background/Context:
Most students with disabilities receive the majority of their instruction in general education classrooms. Yet, general education teachers persistently describe feeling unprepared to academically support students with disabilities in those spaces. Because disabled students are typically excluded from mathematics education research, and because special education researchers typically describe mathematics teaching and learning in ways that are incongruent with ambitious mathematics instruction, there is arguably a lack of guidance for these teachers. In the absence of clear guidance, teachers may turn to the well-established mathematical ability hierarchy, which positions disabled students (among others) as less capable.
Purpose/Objective/Research Question/Focus of Study:
The purpose of this study was to uncover teachers’ talk about the mathematical capabilities of students with (and without) disabilities. Existing coding schemes (perhaps inadvertently) treat teachers’ views as uniform across students despite evidence that teachers hold different views of different students, in part because of the multiple and varied identities that students bring to the classroom. By using an adapted interview protocol, which yielded more (and more nuanced) analytic categories, I foregrounded students’ disability status as a factor that could relate to differences in teachers’ conceptions of who they view as mathematically capable.
Research Design:
I interviewed general education mathematics teachers (N = 20) about their students (n = 407) using an adapted version of Jackson et al.’s (2017) semi-structured protocol that focused on uncovering teachers’ usages of diagnostic and prognostic frames. I used open and concept coding to develop an expanded version of Jackson et al.’s coding scheme and then applied the new coding framework to the entire data set. I used student demographic data to compare within-group percentages, noticing to what degree students with disabilities were represented within particular qualitative categories in relation to their representation within the entire data set. I also used transformed data to estimate two multinomial logistic regressions: one that used diagnostic frames as the outcome variable, and one that used prognostic frames as the outcome variable. Both models used students’ disability status and teacher dummy codes as predictor variables.
Conclusions/Recommendations:
The majority of teachers in this sample explained mathematical struggle in unproductive terms and said they would aim instructional adjustment at unproductive outcomes for students with and without disabilities. However, students with disabilities were overrepresented in unproductive categories and underrepresented in productive categories in relation to both diagnostic and prognostic frames. Regression analyses indicated that a student was statistically less likely to get a productive diagnostic or prognostic frame if they had a disability label. Findings from this study highlight the necessity of including teachers’ views of their students’ mathematical capabilities in instructional improvement efforts. Second, they indicate that student-level factors, such as disability status, relate to qualitatively and quantitatively meaningful differences in teachers’ views of their students, suggesting the importance of attending to broader narratives around constructs that may be associated with teachers’ views, and the subsequent enactment of those views, in mathematics instruction.
Keywords
Mathematics is an essential content area for all students, but especially students with disabilities. 1 For students with disabilities, mathematical skills relate to improved school performance, including postschool outcomes (Benz et al., 2000); beyond traditional views of “success,” these skills also can support disabled students in working toward goals they have as mathematicians and as people (Tan & Kastberg, 2017). In light of the practical and personal importance of learning mathematics, students with disabilities often struggle to demonstrate their mathematical understanding on standardized tests (e.g., National Assessment of Educational Progress [NAEP], 2022) and other conventional measures, prompting description of an “achievement gap” between students with and without disabilities (Wei et al., 2012). Although this significant gap has persisted for nearly three decades (NAEP, 1990–2022), focusing on outcomes—in this case, achievement differences—perhaps promotes static and deficit views based on student characteristics (Gutiérrez, 2008). A more productive way to conceptualize an achievement discrepancy is to interrogate the different opportunities students have to learn (e.g., Kurz et al., 2014) or the “opportunity gap” that exists between students with and without disabilities.
Learning opportunities manifest at the classroom level and are facilitated by the teacher. Although learning opportunities for students with disabilities are typically aimed at mastering mathematical computation and procedures (Foegen & Dougherty, 2017; Lewis & Fisher, 2016), federal policy and recent case law have articulated a set of rigorous learning goals for students with disabilities that go beyond computation and procedures. The Individuals with Disabilities Education Act (2004) and the U.S. Supreme Court’s Endrew F. decision (U.S. Department of Education, 2017) assert that students with disabilities should have the opportunity to meet goals that are both challenging and ambitious. However, the existence of policy or legal standards is not enough to prompt action; the decision to enact learning opportunities aimed at rigorous learning goals is ultimately left to individual teachers. Because most students with disabilities receive the majority of their instruction in general education settings (U.S. Department of Education, 2021), general education teachers are positioned as the daily instructional decision makers for those students. Yet, general education mathematics teachers consistently report feeling unprepared to instructionally support students with disabilities (Maccini & Gagnon, 2006; Perry et al., 2015). This is unsurprising given that students with disabilities are often excluded from mathematics education research in general (Lambert & Tan, 2017) and, more specifically, from talk about mathematical opportunity gaps (e.g., Flores, 2007). General education mathematics teachers, in an effort to gain insight into instructionally supporting students with disabilities, might turn toward special education. However, special education researchers who study the teaching and learning of mathematics often use theoretical framings, methodologies, and epistemologies that are incommensurate with challenging and ambitious mathematics instruction (Labert & Tan, 2017). Taken together, these circumstances arguably indicate a lack of guidance for general education mathematics teachers about how to instructionally support students with disabilities in working toward and achieving rigorous learning outcomes (e.g., DeSimone & Parmar, 2006).
In the absence of professional guidance, general education mathematics teachers are left to interpret educational policy in relation to their existing knowledge and views (Spillane et al., 2006), which could mean filtering policies through the well-established mathematical ability hierarchy in which students with disabilities (among others) are positioned as less capable (Louie, 2017). Teachers’ views about their students’ mathematical capabilities have been linked to the instructional decisions that general education mathematics teachers articulate making (Jackson et al., 2017) or actually make. For example, Wilhelm (2014) investigated the relation between teachers’ views of students perceived as struggling, and classrooms with students of color and emergent bilinguals (Wilhelm et al., 2017). In both studies, teachers’ views were statistically significantly related to whether or not teachers’ employed instructional practices aimed at rigorous mathematical activity. These qualitative and quantitative relations signal that researchers should continue to attend to teachers’ views of their students, given what these views could mean for the mathematical learning opportunities afforded to students, including those with disabilities.
Special education research about disabled students’ mathematics learning has, in some instances, framed instructional decision-making in relation to seemingly “neutral” data, such as diagnostic assessments (e.g., Ketterlin-Geller & Yovanoff, 2009), progress monitoring (e.g., Stecker et al., 2008), and standardized tests (e.g., Hamilton et al., 2009). In these instances, equipped with student data, the teacher continues to be the agent through which instructional opportunities are chosen. However, even when equipped with data, teachers may still make decisions based on their views of students’ capabilities (e.g., Faulkner et al., 2013). This suggests that investigating the role of teachers’ views in the instructional decision-making process could shed light on the opportunity gap that exists between students with and without disabilities, possibly illuminating, in part, a factor that may relate to the persistent opportunity gap and, subsequently, disabled students’ opportunity to learn.
The purpose of this study was to investigate the nuanced ways in which general education mathematics teachers viewed their students’ mathematical capabilities, uncovering whether those views differed between students with and without disabilities. In the following section, I describe two analytical frameworks: the social model of disability, and frame analysis. Within frame analysis, I describe two framing tasks—diagnostic and prognostic framing—and relate them to the construct of teachers’ views of students’ mathematical capabilities.
Models of Disability as Sensemaking Lenses
How a person conceptualizes the construct of disability necessarily informs how they interpret information about disabled people. Because multiple models of disability exist—some are used more by different segments of the educational research community, and some are used less—I unpack a few models and articulate my orientation to disability so that findings shared in this article make sense. (That is, someone with a different orientation to disability would “see” entirely different things in these data, yielding meaningfully different findings.) Two of the most common models of disability are the medical model and the social model. The medical model suggests that disability is the result of inherent biological differences that produce nonnormative mental or physical functioning (Turnbull & Stowe, 2001). Within the medical model, a common response to disability is to aim intervention and remediation at the person in order to fix the abnormality and produce more “normative” functioning (Haegele & Hodge, 2016). In contrast, the social model of disability acknowledges that inherent biological differences exist between people, though the presence of such differences is not inherently disabling. Rather, society’s response to those differences is what disables people (Gallagher et al., 2014). For example, a person with an intellectual disability is only “disabled” by their disability if, in response to their intellectual difference, texts are not created using, or made available through, plain language. In this example, it is not the person or their mind that caused the text to be inaccessible, but the text itself. Within the social model, a common response to disability is to aim intervention and remediation at the environment in order to remove barriers (Oliver, 1996). Whereas the medical model persists within the extant special education research literature (Lambert & Tan, 2017), I used the social model of disability as an interpretive lens through which to understand the ways that teachers in this study framed problems and solutions related to student learning, and to consider the findings in terms of teachers’ agency (and not students’ unresponsiveness to instruction).
Frame Analysis
Goffman (1974) described frames as lenses that individuals use to make sense of what is going on around them. Later conceptualized to describe how people make sense of and organize toward collective political action (e.g., Snow & Benford, 1988), frame analysis has been recently adopted by educational researchers as a way of characterizing teachers’ talk about problems of practice. Three types of frames facilitate this sensemaking process: diagnostic, prognostic, and motivational (Snow & Benford, 1988). Diagnostic frames go beyond identifying a problem and reveal the underlying cause of the problem. For example, a student’s difficulty learning mathematics might be described as a problem. Yet, different teachers might explain the source of that difficulty differently (e.g., attention-deficit hyperactivity disorder or poor instruction). Prognostic frames are concerned with both solutions to the problem and strategies to enact those solutions. For example, a teacher might propose increasing the number of breaks a student could take, or ensuring that the student had access to high-quality tasks to address the problem of the student’s mathematics difficulty. Motivational frames are then about compelling people to act by developing a clear reason for action. Motivational frames were excluded from this study, given their orientation to the collective action of a group. Instead, the current study focuses on teachers’ diagnostic and prognostic frames.
Explanations for Students’ Struggle (Diagnostic Frames)
Jackson and colleagues (2017) used frame analysis and identified two dimensions of teachers’ views of students’ mathematical capabilities: explanations for students’ struggle, and articulated instructional adjustments. Explanations for students’ struggle, or diagnostic frames, are explanations of what teachers identify as the underlying cause of students’ struggle. Understanding the nature of teachers’ explanations for struggle can reveal whether struggle is conceptualized as immutable or malleable, as something that that is located within the student or the environment in which the student is trying to learn. The extant literature reveals that struggle has consistently been framed as something that exists within the student (and is therefore immutable), whether in terms of innate ability (Westwood, 1995), mathematical talent (Baştürk, 2016), psychological causes (Bingolbali et al., 2011), or characteristics of the student, such as laziness (Jackson et al., 2017). Some general education mathematics teachers also cited students’ disabilities as the primary cause of their mathematical struggle (Shifrer, 2016). Uncovering teachers’ explanations for students’ struggles likely has implications for the instructional opportunities afforded to students. “If teachers fail to accept responsibility for students’ successes or failures and thus fail to see a relationship between their behavior and students’ performance, they would be less likely to work to improve their students’ performance in the classroom” (Clark & Peterson, 1984, p. 93). That is, teachers who view their students as mathematically capable, and thus explain students’ struggles in terms of the learning environment or instructional opportunity, may be more likely to take instructional responsibility for supporting students in achieving rigorous learning outcomes.
Articulated Instructional Adjustments (Prognostic Frames)
The instructional adjustments that teachers say they would make for students experiencing mathematical struggle, or prognostic frames, identify particular instructional supports that allow students to participate in rigorous mathematical activity. Understanding the nature of instructional adjustments could provide insight into the types of learning opportunities students with disabilities might have, as well as reveal the degree to which general education teachers have the appropriate resources to support students with disabilities in working toward and achieving rigorous mathematical outcomes. For students with disabilities, the nature of instructional adjustments typically de-emphasizes the importance of academic learning, instead emphasizing the social, emotional, or behavioral benefits of instruction (Cameron & Cook, 2013). Instructional adjustments for students with disabilities often involve decreasing the rigor of mathematical activity by allowing students to “opt out” of completing challenging tasks (Mayrowetz, 2009); by minimizing opportunities for mathematical discourse (Wiebe Berry & Kim, 2008); or by avoiding specific mathematical content perceived to be complex (e.g., algebra, word problems; DeSimone & Parmar, 2006).
Although the nature of instructional adjustments has been documented, what remains unexamined are teachers’ articulated rationales for the instructional adjustments they say they would make. That teachers might name the same instructional adjustment but have different rationales (that differ in quality; Saxe et al., 1999) suggests that hearing teachers articulate their reasoning for making an instructional adjustment would give more nuanced insight into the instructional decision-making process. For example, knowing that a teacher might name “using manipulatives” as a support does not immediately reveal a teacher’s thinking about what they see as the goals of using manipulatives. Based only on existing mathematics education and special education literature, the opportunity for students to use manipulatives would be considered generally productive. Yet, without clarity about the teacher’s purpose for presenting the student with manipulatives, determining the nature of an instructional adjustment is potentially speculative.
In light of the role teachers that have in shaping the instructional opportunities afforded to students with disabilities, the current study was driven by the following research questions:
How do general education mathematics teachers explain students’ struggle?
What is the nature of the rationales teachers give for instructional adjustments?
What is the relation between students’ disability status and the nature of teachers’ explanations and rationales?
Method
I used a mixed methods design to address the research questions posed in this study. The purpose of using mixed methods was to enhance the credibility of the findings by examining to what degree results from the qualitative and quantitative analyses could be triangulated (Greene, 2007). In this study, the primary data set consisted of qualitative interview data, with a secondary data set consisting of two types of quantitative data: transformed qualitative codes and categorical demographic data. Qualitative data were collected and analyzed before data transformation and quantitative analysis. Interpretation involved comparing qualitative and quantitative findings to determine the degree to which findings converged.
Sampling and Teacher Demographics
After receiving institutional review board approval, teacher participants were recruited from two public midwestern (U.S.) school districts. This study included 20 general education mathematics teachers. The majority of the teachers were recruited from a large suburban school district, District A, that comprised approximately 10,500 students, 6% of whom were identified as students with disabilities. The remaining teachers were recruited from another large suburban school district, District B, and comprised approximately 17,800 students, of whom 4.5% were considered students with disabilities. Elementary and middle grades (i.e., K–8) teachers were targeted for participation given the likelihood that students in those grade levels would remain enrolled in a yearlong mathematics course. This yearlong enrollment ensured that teachers had ample time to develop informed views about their students before participating in the current study. I used purposive criterion sampling (Gall et al., 2007) and included interested teachers if they (a) held a current teaching license and certificate, (b) taught mathematics, and (c) had at least one student with an identified disability in at least one of their classes.
At the time of the interview, teachers were asked to read and sign a consent form. Table 1 lists teacher demographics, including their school district, total years of teaching experience (including the 2018–2019 school year), the grade level they were teaching at the time of the interview, their highest degree earned, and their observed race and gender. The majority of teachers in this study were identified as female and White, taught students in Grades 7 and 8, had 10 or more years of teaching experience, and held a master’s or other advanced degree. At the end of each interview, teachers were compensated with a small gift card.
Teacher Demographics and Students With Disabilities Within Teachers’ Classrooms.
Note. BS = bachelor of science; Elem. = elementary; Ed. = education; MEd = master of education; Admin. = administration; MBA = master of business administration; Psych. = psychology; SpEd = special education; MAT = master of arts in teaching; EdD = doctor of education; EdS = education specialist; HR = human resources.
Transcript was included in coding scheme development.
Student Demographics
Each teacher was asked to bring the student roster of one of their classes to the interview; if complete qualitative data were reported during the interview (i.e., data that yielded both a diagnostic and prognostic code), students’ data were then included in the current study. Across the 20 teacher participants, there were complete qualitative data and disability status demographics for 407 students. Table 1 shows the number of students with complete qualitative data from each teacher’s class, and the number of students with disabilities within each class. A total of 26 students were excluded from this data set for the following reasons: Students were new to the classroom, and teachers did not know them well enough to answer interview questions about them (n = 4); missing diagnostic framing (n = 2); missing prognostic framing (n = 8); and missing demographic data (n = 12). Approximately 30% (n = 121) of the total student data set consisted of students with identified disabilities, including other health impairment (n = 34), learning disability (n = 24), emotional disturbance (n = 15), speech or language impairment (n = 13), autism (n = 9), intellectual disability (n = 3), hearing impairment (n = 1), traumatic brain injury (n = 1), multiple disabilities (n = 1), or not specified (n = 20).
Instrument Development
The interview protocol used in this study was adapted from two existing interview protocols (see Jackson et al., 2017, and Munter, 2014). Before this study, the adapted interview protocol was vetted by experts in special education and mathematics education and pilot tested with four middle-grade (i.e., Grades 6–8) general education mathematics teachers from a midwestern (U.S.) school district separate from the districts that participated in the study. As a result of consultation and pilot testing, I made minor wording changes to the interview protocol.
Qualitative Data Collection Procedures
Qualitative data collection occurred in one phase, with data reduction, transformation, and analyses following. In-person, semi-structured interviews (Merriam & Tisdell, 2016) were conducted individually between February and June 2019. Each interview (N = 20) lasted approximately 90 minutes (ranging in length between 55 and 130 minutes) and was audio recorded. Teachers were asked to select one of their classes that included at least one student with an identified disability and bring a paper roster of students’ names to the interview. Following the interview protocol, after answering questions about their vision of high-quality mathematics instruction (Munter, 2014), teachers were asked to cut the paper roster into strips so that each student’s name was on one strip of paper. Teachers were then asked to physically manipulate the pieces of paper to create a rank-ordered list of students, organized from “most struggling” to “least struggling,” however they defined struggling. Once teachers completed this task, they were asked to articulate how they defined struggling. For the remainder of the interview, teachers answered the same two questions about each student listed on the roster, which I describe in the “Qualitative Coding Scheme Development” subsection.
Data organization and reduction
Each audio-recorded interview was transcribed. I listened to the audio recording and transcribed it verbatim using a researcher-developed transcription key. To organize the primary data set into analytic units, stanzas were created within each transcript. A stanza was defined as a sentence or group of sentences that represented a cohesive idea or topic (Gee, 2014). In addition to analytic benefits, creation of stanzas allowed data that were beyond the scope of this analysis or otherwise unrelated to this research to be easily identified and omitted. The final primary qualitative data set was derived from approximately 1,690 minutes of audio recordings and included 790 stanzas (63% of the complete data set), generated from 20 teachers, and teachers’ talk about their 407 students.
Qualitative Coding Scheme Development
The Views of Students’ Mathematical Capabilities-Expanded (VSMC-E) coding scheme was developed to capture nuances in teachers’ talk about their views of individual students in terms of both the source of student struggle and the rationale teachers gave for making instructional adjustments. In their study, Jackson et al. (2017) asked the question, “When your students don’t learn as expected, what do you find are typically the reasons?” (p. 9). Teachers’ responses were then interpreted in relation to “students” and coded as productive, unproductive, and mixed—categories that served as provisional codes (Saldaña, 2021) for the VSMC-E. However, because I speculated that teachers would articulate different sources and adjustments for different students, I adapted the question to read, “When [this individual student] doesn’t learn as expected, what do you find are typically the reasons?” Similarly, after asking about the instructional adjustments the teacher said they would make for a student, I adapted Jackson et al.’s second question to be, “How does [the instructional adjustment named] help that student?” Using the student roster, teachers responded to these questions about each of their students, which yielded qualitatively more specific and nuanced codes, thus expanding the original scheme.
To develop the VSMC-E, I divided the qualitative data into two subsets, a development data set and a reliability data set. The development data set consisted of a representative 20% of the transcripts (n = 8 transcripts; see Table 1 for details about which transcripts were included in the development data set). The development data set was considered representative because the transcripts were from teachers who proportionally represented the whole data set based on demographic characteristics such as years of teaching experience, race, gender, and school district. I read each stanza within the development data set and initially assigned a provisional code. Then, looking within each provisional code, I identified patterns that were then categorized into additional codes. Once all codes were identified, I conducted concept coding (Saldaña, 2021), with the goal of identifying disconfirming evidence (Brantlinger et al., 2005). As instances of disconfirming evidence surfaced, I examined the data in relation to the development data set, and instances were absorbed into existing codes, resulted in the creation of a new code, or resulted in redefining an existing code.
Next, to confirm and clarify the VSMC-E codes, I conducted intercoder agreement, in which coders work together to arrive at agreement about the appropriate code for a particular stanza (Campbell et al., 2013). Two additional researchers participated in the coding agreement and reliability processes. Researcher 1 had a PhD in mathematics education, was an experienced coder, and participated in the development of the diagnostic framing portion of the coding scheme; Researcher 2 had a PhD in special education, was an experienced coder, and participated in the development of the prognostic framing portion of the coding scheme. Both researchers were provided one 2-hour training in which we read through a researcher-created coding guide. Because intercoder agreement is especially useful when conducting exploratory research (Campbell et al., 2013), which can include coding scheme development, both researchers and I compared codes; when there were discrepancies, we talked about our individual rationales for assigning each code and came to a final agreement. This process resulted in the final VSMC-E coding scheme, which comprised seven diagnostic codes (i.e., explanations for students’ struggle) and seven prognostic codes (i.e., rationales for instructional adjustments).
To establish intercoder reliability, both researchers and I independently coded 22.55% of the students represented in the development data set (i.e., 92 students from eight teachers). For diagnostic framing, final intercoder agreement between me and Researcher 1 was 70.65%, with a kappa of .56, indicating moderate agreement (Landis & Koch, 1977). For prognostic framing, final intercoder agreement between me and Researcher 2 was 75.76%, with a kappa of .68, indicating substantial agreement (Landis & Koch, 1977). While the goal was to achieve substantial agreement for both diagnostic and prognostic framing, the relative complexity of this coding scheme suggests that arriving at substantial agreement (or “almost perfect” agreement, Landis & Koch, 1977, p. 165) requires additional coding scheme refinement (see Campbell et al., 2013, for a description of the difficulty of coding in-depth semi-structured interviews).
Qualitative Data Analysis
After establishing the reliability of the coding scheme and to address the first and second research questions, I used the VSMC-E codes to code the remainder of the primary data set (n = 12 transcripts) and any uncoded stanzas from the development data set. Each student within each teacher’s class was assigned one diagnostic code and one prognostic code, resulting in two qualitative codes per student.
Quantitative Data Collection Procedures
Quantitative data collection occurred in one phase, with data reduction, transformation, and analysis following.
Quantitative Data Reduction and Transformation
The secondary data set consisted of two types of quantitative data: transformed qualitative codes and categorical demographic data. To transform qualitative data into quantitative data, each qualitative VSMC-E code was assigned a corresponding numeric value.
There were two outcome variables in the current study, diagnostic frames and prognostic frames. For qualitative analysis, I used all seven diagnostic and prognostic VSMC-E codes; to ensure that data were suitable for quantitative analysis, I reduced the number of codes from seven to three categories: unproductive, mixed, and productive. I made this reduction given the recommendation that there be at least 10 observations per predictor variable for each outcome category (Pituch & Stevens, 2016). In the current study, the predictor variable within both statistical models was student disability status. When examining the disaggregated data across the seven qualitative categories, the recommendation for 10 students with disabilities within each outcome category was not met. Therefore, reducing the seven categories to three categories allowed for this recommendation to be satisfied. Each participating school district provided student-level demographic data. I used the teachers’ names and students’ initials to match demographic information with qualitative codes. The categorical variable of interest, student disability status, was dummy coded.
Next, there were three teachers in the sample whose classroom data were not suitable for quantitative analysis. All three teachers had two or fewer students with disabilities in their respective classes. In addition, during their interviews, each teacher explained they did not think about instruction at the individual student level and therefore gave responses that characterized small groups of students within their classes; this resulted in the assignment of identical qualitative coding patterns across the small groups. Taken together, these factors suppressed needed variability in the data set and were removed before quantitative analysis. Therefore, the quantitative data set consisted of 17 teachers and 336 students (82% of the original data set).
Given the nested nature of these data (e.g., students in the same class, with the same teacher), teacher-level characteristics must be acknowledged. Although specific teacher characteristics (e.g., years of experience teaching, specific content area knowledge, self-efficacy) may be related to teachers’ views of students’ mathematical capabilities, the focus of this analysis was on the relation between student-level characteristics and teachers’ views. Therefore, teacher-level characteristics were considered fixed effects; thus, differences in individual teachers were acknowledged but controlled for by creating one dummy code per teacher.
Quantitative Data Analysis
To address the third research question, descriptive statistics of within-group characteristics were analyzed as percentages, noticing whether students with or without disabilities were represented within certain qualitative categories in larger or smaller proportions relative to their representation within the entire data set. Next, I used the quantitative data set to conduct two multinomial logistic regressions. I estimated two regressions, one that used diagnostic frames as the outcome variable, and one that used prognostic frames as the outcome variable. Both outcome variables included three categorical levels: unproductive, mixed, and productive. Both models used students’ disability status and teacher dummy codes as predictor variables. The chi-square statistic was analyzed to determine the significance of the model, then coefficients and odds ratios were calculated. Results were analyzed through coefficients and odds ratios that indicated whether or not disability status made a statistically significant (p < .05) contribution to the likelihood of a certain view about students’ mathematical capabilities.
Findings from Qualitative Analyses
The development of the VSMC-E coding scheme served as an analytical tool but was also considered a finding. Details about the nature of each qualitative code are shared first, followed by a description of how those codes were distributed in this sample, and, finally, the outcomes of quantitative analysis.
How Do General Education Mathematics Teachers Explain Students’ Struggle?
To determine teachers’ diagnostic frames (i.e., explanations for struggle), teachers were asked, “When [this individual student] doesn’t learn as expected, what do you find are typically the reasons?” We used seven codes to characterize teachers’ responses; Table 2 describes the codes and gives an example from the data. We coded responses as unproductive inherent if the teacher explained student struggle in terms of inherent qualities of the student or deficit notions of the student’s community or family (e.g., being academically “low”; parents do not value education). We coded responses as unproductive behavior/attitude if the teacher explained student struggle in terms of temporary or malleable student behaviors (e.g., rushing to complete work). We coded responses as productive circumstance if the teacher explained student struggle in terms of circumstances outside the teacher or school’s control (e.g., frequent absences) but without placing blame on either the student or their family. We coded responses as productive instructional if teachers explained student struggle in terms of instructional or school-based opportunities (or the lack of) (e.g., student did not have an opportunity to learn in a previous grade). We coded responses as mixed if the teacher explained student struggle in terms of a combination of either productive code and either unproductive code (e.g., student did not have an opportunity to learn in a previous grade and a student’s bad attitude). We coded responses as not applicable if the teacher explained student performance as “expected” and unknown if the teacher said they did not know why a student would struggle to learn mathematics.
Views of Students’ Mathematics Capabilities-Expanded (VSMC-E) Coding Scheme.
What Rationales Did Teachers Give for Instructional Adjustments?
To determine teachers’ prognostic frames (i.e., rationales for instructional adjustments), they were asked to name any instructional adjustments they made for an individual student. Then, teachers were asked, “How does [the instructional adjustment named] help that student?” (see Table 2 for examples). We coded responses as unproductive behavior/self-esteem if the teacher described making an instructional adjustment aimed at managing behavior or toward a behavioral outcome (e.g., keeping a student “on task”), or the teacher described making an instructional adjustment aimed at students’ confidence or self-esteem (e.g., so a student felt better about themselves). We considered and coded these responses together, given that neither had a learning orientation. We coded responses as unproductive learning if the teacher described making an instructional adjustment aimed at decreasing the rigor of the task or learning outcome (e.g., solving problems procedurally). We coded responses as productive access if the teacher described making an instructional adjustment aimed at giving a student access to a learning opportunity by altering some aspect of the instructional context (e.g., reading problems aloud). We coded responses as productive learning if teachers described making an instructional adjustment aimed at rigorous outcomes (e.g., developing conceptual understanding). We coded responses as mixed if the teacher described making an instructional adjustment aimed at a combination of either productive code and either unproductive code (e.g., supporting a student in developing understanding of a mathematical concept and keeping a student “on task”). We coded responses as not applicable if the teacher described not making an instructional adjustment for a student because either the current instruction was “working” for the student, or the student did not require additional instructional adjustments. We coded responses as ineffective if a teacher articulated they did not make instructional adjustments because they perceived that the adjustment would not help the student.
To specifically address Research Questions 1 and 2, the VSMC-E coding scheme was applied to the entire data set; Table 3 shows the distribution of teachers’ responses about students across the seven diagnostic and prognostic frames. For the majority of students (n = 250; 61.43%), general education mathematics teachers explained students’ struggle in unproductive terms. That is, teachers in this study typically named things that students did or were as an explanation for students’ struggle in mathematics. In relation to prognostic frames, teachers said they would aim the majority of instructional adjustments (n = 257; 63.14%) at unproductive outcomes. Here, teachers described making instructional adjustments that were diminished in quality or that were oriented toward managing students’ behavior or promoting the development of confidence or self-esteem (without attending to student learning).
Teachers’ Explanations for Struggle and Rationales for Instructional Adjustments.
Note. Unknow. = unknown; Unpro. = unproductive; Pro. Circum. = productive circumstance; Pro. Instruct. = productive instructional; N/A = not applicable; Ineffect. = ineffective; Pro. = productive; SWD = students with disabilities; SWoD = students without disabilities. Percentages in the “All” row should be interpreted in relation to the entire data set (N = 407); percentages reported in the “SWD” and “SWoD” rows should be interpreted in relation to each category.
Findings from Quantitative Analysis
When examining the full data set, students with disabilities (n = 121) comprised 29.73% of the total data set. Table 3 also includes comparisons between students with and without disabilities across all diagnostic and prognostic frames. When comparing teachers’ diagnostic frames for students with and without disabilities, students with disabilities were overrepresented in the unproductive inherent category (n = 67; 51.15%) and underrepresented in the productive instructional category (n = 6; 18.75%). When comparing teachers’ prognostic frames for students with and without disabilities, students with disabilities were overrepresented in terms of students for whom instructional adjustments were aimed at unproductive behavior outcomes (n = 49; 36.30%) and underrepresented in terms of students for whom instructional adjustments were aimed at productive learning outcomes (n = 6; 21.43%).
The results of a multinomial logistic regression indicated that disability status added significantly to the overall model and, in some instances, was a significant predictor. Using the reduced quantitative data set, when estimating the diagnostic and prognostic models with all students (N = 336), and without specifying disability status, the diagnostic model yielded a nonsignificant chi-square statistic, X2 = 412.81, p > .05, and the prognostic model yielded a significant chi-square statistic, X2 = 461.36, p < .05. After adding students’ disability status into the models, the chi-square statistic became significant, X2 = 70.19, p < .001, and more significant, X2 = 113.69, p < .001, respectively, which indicated that students’ disability status explained a significant amount of the original variability and was a better fit than the original models.
Table 4 displays the outcomes of both regression models. In terms of diagnostic frames, disability status significantly predicted whether a student’s struggle was explained in productive terms, b = -.81, p < .05, OR = .44. The odds ratio means that students were significantly less likely to get a productive explanation than an unproductive explanation if the student was labeled with a disability. In terms of prognostic frames, disability status significantly predicted whether the instructional adjustments were aimed at productive outcomes, b = -1.30, p < 0, OR = .27. The odds ratio means that students were significantly less likely to get a productive rationale than an unproductive rationale if the student was labeled with a disability.
Odds Ratios for Diagnostic and Prognostic Frames.
p < .05. **p < 0.
Discussion
Outcomes from this mixed methods study revealed two main findings. First, based on their framings, teachers in this sample did not tend to view students with disabilities as mathematically capable, and second, there were meaningful differences between teachers’ views of students with and without disabilities. Triangulated data support these findings.
The general unproductive nature of teachers’ views about their students is consistent with other empirical studies and raises questions about the degree to which teachers may or may not feel they can instructionally support students experiencing struggle. Diagnostically, finding that general education teachers attributed the struggle of students with disabilities to internal stable characteristics, such as disability, echoes other scholars’ observations (e.g., Harry & Klingner, 2007; Jordan et al., 2009). Evidence continues to indicate the relation between teachers’ explanations for students’ struggle, and the likelihood that teachers will enact mathematical practices that vary in quality (Wilhelm, 2014; Wilhelm et al., 2017). The relation between teachers’ views and enacted instructional practice is of practical importance, given the role that general education teachers have in shaping the type of learning opportunities afforded to students with disabilities, and the outcomes of those learning opportunities.
Prognostically, unproductive behavior frames are considered less productive than unproductive learning frames. I contend that while instruction aimed at decreasing the rigor of mathematical activity is not ideal, it is preferred over instruction aimed at nonlearning outcomes. Teachers in the current study often identified instructional adjustments that themselves could be considered instructionally productive. However, in multiple instances within the data set, teachers articulated nonacademic reasons for making the adjustment. For example, teachers in the current study named “giving students manipulatives” as an instructional adjustment they would make, but articulated reasons for making such an adjustment were aimed at behavioral outcomes (e.g., keeping a student engaged, preventing a student from touching peers). The special education and mathematics education literature bases identify the use of manipulatives as an evidence-based practice that supports a range of students, including those with disabilities, in reasoning about and solving mathematical problems (e.g., National Council of Teachers of Mathematics, 2014; Powell & Fuchs, 2015). Yet, in this and other studies, general education teachers de-emphasized the importance of academic learning for students with disabilities, emphasizing instead the social, emotional, and behavioral aspects of learning (e.g., Cameron & Cook, 2013). This de-emphasis on academic learning points to another factor that may relate to the mathematical opportunity gap between students with and without disabilities and further highlights the need to interrogate the rationales teachers have for making instructional adjustments beyond simply identifying what those adjustments are.
The quantitative and qualitative findings are strengthened through triangulation (Brantlinger et al., 2005; Greene, 2007), or the use of multiple methods to investigate one topic. In terms of methodology, there have been calls for the increased use of mixed methods research in special education as a way to expand the range of research questions that are being asked and as a way to gain a better understanding of the complexities of the field (Connor et al., 2011; Corr et al., 2020). Empirically, both sets of data in the current study reinforce the overarching finding: When disability status was accounted for, there were qualitative and quantitative differences in teachers’ diagnostic and prognostic frames, which suggests that disability status is related to teachers’ views of their students’ mathematical capabilities.
Limitations
The VSMC-E coding scheme was a first attempt at adding qualitative nuance to the original coding scheme and has not been tested by other researchers or used in other contexts. Similar to any analytic tool, qualitative coding schemes are not intended to capture all possible aspects of any one construct (Saldaña, 2021). However, a noteworthy limitation of the VSMC-E was its inability to account for the positive or supportive talk that teachers engaged in when describing their students. While the focus of the current study was on teachers’ explanations for students’ struggle, that teachers did articulate students’ assets suggests that a secondary coding scheme could be developed, one aimed specifically at characterizing teachers’ talk about what they perceive facilitates students’ mathematical success. Relatedly, intercoder agreement is an area of growth, specifically the diagnostic framing portion of the scheme. Because intercoder agreement is especially difficult when coding semi-structured interviews (Campbell et al., 2013), additional attention should be paid to further developing this scheme toward categories that represent perhaps a degree of stability.
In terms of generalizability, the teacher sample included in this study was demographically homogenous, and the majority of teachers came from one school district, primarily at the middle school level (i.e., Grades 6–8). These limitations may reflect local or contextual factors and, if conducted with different teachers, may result in different findings. Similarly, the student sample included in this study reflected a greater proportion of students with disabilities (29.73%) than is nationally representative (13.7%; McFarland et al., 2019). This overrepresentation may again reflect local or contextual factors that, in future investigations, should be accounted for or leveraged.
Implications for Research
The current study adds nuance to our understanding of teachers’ views of students’ mathematical capabilities by using an adapted interview protocol and expanding an existing coding scheme, yet iterations of this study are needed to validate these tools. Iterations of this study could benefit from a larger sample size, which would provide additional opportunities to test whether the VSMC-E codes are applicable to a range of mathematics teachers’ responses. A larger sample would also provide opportunities to qualitatively and quantitatively investigate additional differences that may exist based on specific disability categories. In the current data set, 10 of the 13 federally recognized disability categories were represented. One might speculate that different disability labels would be more or less related to different framings. Although these stereotypical views did occur in the current data set, they did not occur with enough frequency to constitute a qualitative pattern or to allow for quantitative analyses.
Once the VSMC-E is validated, an extension of this study would be the coupling of the VSMC-E with classroom observations. The original Views of Students’ Mathematical Capabilities protocol has been used in relation to classroom observations and enacted instructional practices (e.g., Wilhelm, 2014; Wilhelm et al., 2017). Although the purpose of the present study was to characterize teachers’ talk in more nuanced ways, empirically studying the relations that may exist between teachers’ articulated talk about students with disabilities and enacted instructional practices for those same students could inform the field’s understanding of the mathematical opportunity gap that exists between students with and without disabilities.
Implications for Practice
The underlying purpose of this study was not to characterize the participating teachers in deficit terms or point out their instructional shortcomings. Many teachers in this study held complex views of their students and in many instances knew their students as multidimensional people. Rather, these data suggest that general education mathematics teachers are underresourced in enacting instructional practices aimed at supporting a range of students in rigorous mathematical activity. One resource that might be tailored to address this particular problem is a reconceptualization of the learning opportunities afforded to in-service teachers. Multiple studies suggest that when teachers see students they perceive as struggling participate in and benefit from a particular mathematical instructional practice, they experience a shift in their views of those students (e.g., Darragh & Valoyes-Chávez, 2019; Guskey, 2002; Jackson et al., 2017), which may then relate to future enacted practice. Professional learning opportunities for in-service teachers could then be structured to promote the use of practices that are designed toward rigorous mathematical activity, coupled with ongoing reflection and discourse around teachers’ shifting views of disability and capability (e.g., Beneke et al., 2022).
General education mathematics teachers also likely require more robust training in relation to supporting a range of students in participating in such rigorous mathematical activity. General education mathematics teachers consistently report feeling instructionally unprepared to support students with disabilities in general education settings (Maccini & Gagnon, 2006; Perry et al., 2015). This feeling is reflected by the teachers in the present study. Though not the primary focus of the current study, all teacher participants reported that they had never had professional development specifically aimed at supporting students with disabilities in mathematics. Given these findings, it seems necessary for teachers to be supported in both enacting ambitious mathematical practices and doing so for a range of students. A first step might be for teachers to articulate their views of disability and mathematics capability. In relation to disability, scholars have found that pre- and in-service teachers may articulate productive views about the idea of working with disabled students, but further investigation reveals that either those teachers do not want disabled students placed in their classrooms (Hwang & Evans, 2011), or, despite productive talk, they retain deficit language and framings (Mason & Connor, 2022). These findings point to the limitations of simply asking teachers for their views. Beyond allowing teachers to state what their views are, professional development efforts can give teachers the opportunity to interrogate how those views were formed and what reinforces those views in the present. Further, interdisciplinary professional learning opportunities (e.g., Mason & Thomas, 2021) might support both general and special education teachers in developing robust instructional practices that are grounded in content while simultaneously supporting access to rigorous mathematical learning opportunities.
Conclusion
Because general education mathematics teachers are the daily instructional decision makers for students with disabilities, understanding teachers’ views of those students’ mathematical capabilities sheds light on the factors that may relate to teachers’ instructional decision-making. The findings from this mixed methods study suggest that, for this group of teachers, students with disabilities were not viewed as capable of participating in rigorous mathematical activity. This study adds to the existing literature base by extending what was previously known about the ways in which teachers characterize students’ struggle and the rationales teachers give for instructional adjustments. Specifically, this study suggests that student demographic factors, like disability status, relate to differential teacher talk. This work provides initial evidence that students’ characteristics are an important factor in understanding teachers’ views of their students’ mathematical capabilities.
Footnotes
Acknowledgements
I would like to thank Erica Lembke, Chuck Munter, Kara Jackson, and Katie Lewis for their feedback and support throughout the development of this coding scheme and manuscript.
Declaration of Conflicting Interests
The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
The author received no financial support for the research, authorship, and/or publication of this article.