Uncertain Causality Analysis of Critical Success Factors of Special Education Mathematics Teaching

2.1 Self-efficacy

SE refers to the teacher’s inherent capability to organize and implement activities necessary to accomplish their required tasks (Perera & John, 2020; Thurm & Barzel, 2020). In mathematics teaching and learning, such activities include implementing pedagogies and involving the students. Furthermore, the prominence of the notion of SE in the literature is attributed to the increasing focus on certain initiatives such as (a) enhancing job satisfaction, engagement, and commitment in teachers to reduce rates of teacher attrition (McIlveen et al., 2019), and (b) maximizing teacher effectiveness, typically conceived as the effects of in-class teacher beliefs and behaviors on student achievement (Klassen & Tze, 2014). Furthermore, increasing evidence in the literature has shown that teacher SE is linked to improved psychological well-being in terms of higher levels of job satisfaction and commitment and lower levels of stress and burnout (Aloe, Amo, & Shanahan, 2014). In SPED, teachers are more exposed to burnout and stress; however, teachers with high SE feel rewarded during instruction, thus minimizing the risk of burnout (Sarıçam & Sakız, 2014). Moreover, teachers with high SE are less critical of the errors of students, tend to persevere with students who are having difficulties, have more positive classroom management strategies, and are more inclined to the perceived placement of students with disabilities in regular classrooms as appropriate and are less likely to refer these students to SPED (Leyser, Zeiger, & Romi, 2011). Thus, the SE of teachers is an essential aspect of SPED.

2.2 Teaching knowledge

Strong conventional knowledge of the subject matter is essential to make the content topics easily comprehensible to the students (Meyer & Wilkerson, 2011). Since SPED and mathematics require coherent knowledge to make the subject matter understandable, various studies in the literature (e.g., Baker et al., 2022; Sheppard & Wieman, 2020) have explored the significance of TK to the success of teaching in SPED. Accordingly, it is necessary to evaluate the knowledge and expertise of the teacher in terms of (a) mathematics content knowledge, (b) knowledge of how students learn mathematics, (c) knowledge of individual students, and (d) teaching experience (Sheppard & Wieman, 2020). Additional information on the three types of mathematical knowledge (i.e., common content, specialized content, and mathematical horizon) and three types of pedagogical content knowledge (i.e., content and students, content and teaching, and curriculum) identified by Ball, Thames, and Phelps (2008) is essential to understand better the construct of teaching mathematics. Chapman (2013) investigated the effect of a teacher’s content knowledge on their teaching method. He found that teachers with superior mathematical knowledge could write more and better execute specific tasks for their students. Hence, TK is a CSF in SPED (Byrd & Alexander, 2020). Additionally, the teachers must have two core knowledge areas: data collection and assessment of the data collected (Rock et al., 2016). Thus, exploring the causal effect of a teacher’s content knowledge on their teaching practices is crucial in SPED.

2.3 Student feedback

Students’ feedback to a teacher is considered the most underappreciated factor in educational development initiatives (Barana, Marchisio, & Sacchet, 2021; Gormally, Evans, & Brickman, 2014). Feedback from the students, parents, and colleagues can empower and motivate teachers, making them more likely to improve their teaching practices significantly (Henderson, Beach, & Finkelstein, 2011). In SPED, a feedback system is critical in developing teaching pedagogies and initiatives for PD to cater to the needs identified by the feedback (Schles & Robertson, 2019). However, using feedback correctly to assess teaching quality is crucial because it may pose several problems, considering it may be prone to bias and misjudgments (Scherer & Gustafsson, 2015). Nevertheless, there is emerging recognition from the institutions of comprehensive and constructive feedback on the teacher’s instructional practices (Gormally et al., 2014). Through a comprehensive review of teacher education and workplace performance studies, Gormally et al. (2014) emphasized that effective feedback (a) clarifies the task by providing instruction and correction, (b) improves motivation that can prompt increased effort, and (c) is perceived as valuable by the recipient when provided by credible sources.

2.4 Individual factors

Teachers’ individual qualities play a critical role in molding teachers’ propensity to analyze instruction and ultimately influence teacher learning (i.e., degree of integrated knowledge and practice demonstrated), which is vital in SPED (Brownell et al., 2014). Various studies in the literature (e.g., Chan & Yuen, 2014; Chang & Beilock, 2016) have explored how and what specific IF of teachers are critical in improving students’ ability to learn faster. An empirical insight concerning gender and different teaching methods was found by Umugiraneza et al. (2017), wherein males are typically more inclined to use various methods for teaching than females. Also, they found that teachers over 40 are more likely to utilize various assessment strategies compared to teachers younger than 40. Furthermore, teachers with bachelor’s degree qualifications are more likely to use multiple teaching methods than those with postgraduate education. These IF are also emphasized by Kafyulilo et al. (2015) to significantly affect the implementation of technological innovation in a teacher’s instructional practices. Moreover, these IF (i.e., gender, years of experience) are also significantly linked to the teacher’s SE, wherein male teachers and teachers with at least 23 years of experience have better classroom management SE than others (Klassen & Chiu, 2010).

2.5 Professional development

PD is a term used to describe organized, professional learning that enhances teacher effectiveness and student learning outcomes (Darling-Hammond, Hyler, & Gardner, 2017). Kini and Podolsky (2016) emphasized that teachers who enter the professional teaching tier have met a competency standard from which they can expand their expertise throughout their careers and effectively enhance their teaching methodologies. PD includes the teachers’ training to equip them with appropriate skills and competence for instruction (Brock & Carter, 2015). According to observations of SPED programs, inadequate PD prevents teachers from implementing many evidence-based strategies, sometimes with unreasonable loyalty toward traditional pedagogies (Odom et al., 2015). According to Podolsky, Kini, and Darling-Hammond (2019), recent methodological procedures have allowed researchers to match student data with individual teachers and track their effectiveness throughout their careers. Such a procedure allows a closer look at the effect of teaching experience on student outcomes for the same teacher over time as they gain more experience and expertise through training and development. Consequently, Umugiraneza et al. (2017) revealed that teachers who engage in professional learning (i.e., mathematics or statistics workshops) are more likely to utilize multiple assessment strategies for teaching than those who do not.

2.6 Institutional support

The results of Ismail et al. (2015) imply the need for teachers to support high levels of professional community and accountability in their respective departments (e.g., the Mathematics department) and other departments. Various studies in the literature (e.g., Basbeth, Saufi, & Sudharmin, 2021; Seeland, Cliplef, Munn, & Dedrick, 2022) have explored the importance of IS in augmenting any initiative for developing both the skills of students and teachers. Accordingly, there is a need for the whole institution to support the instructional endeavors of teachers to gain more expertise and resources while also minimizing unintentional errors in their teaching instructions (Seeland et al., 2022). In SPED, various studies (e.g., Gomez-Navarro, 2020) have claimed that personal, cultural, and institutional experiences, constituting IS, significantly impact the success of learning students with disabilities. Four types of organizational support were identified by Kraft et al. (2015), namely, (a) educational resources, (b) order and regulative systems, (c) emotional and social support for students, and (d) parent engagement practices. On the other hand, Smith, Booker, Hochberg, and Desimone (2018) revealed that collaboration efforts between teachers, supported by their institutions, do not significantly affect teaching support and quality. Such an observation may be due to the content of the teacher’s interaction not focusing intensely on the subject matter (i.e., mathematics) but on the logistics of instructional resources and educational management. Although various studies emphasize the importance of IS to improve the quality of instruction (e.g., Courduff & Moktari, 2022; Kraft et al., 2015), initiating an efficient framework for the support to acquire meaningful results is crucial, especially in the context of SPED.

2.7 Teaching capacity

One of the main factors that influence the effectiveness of students’ learning outcomes in mathematics is the capacity of teachers, mainly relating to teachers’ knowledge, beliefs, and understanding (Ismail et al., 2015). In SPED, TC pertains to the teacher’s capability or power to purposefully act, take steps, and strategically engage in improvement efforts by actively making daily decisions on inclusive pedagogies that are situation- and context-appropriate (Li & Ruppar, 2021). The initial training on instructional practices influences the capability of the teachers to teach mathematics. Furthermore, their capacity for teaching is influenced by various factors, including their interaction with the available resources, both traditional and digital learning materials (Pepin, Gueudet, & Trouche, 2017). Such interaction combines two interrelated processes: (a) the process of instrumentation and (b) the process of instrumentalization. Instrumentation refers to the influence of the teacher’s knowledge, goals, and values on the characteristics of educational materials. In contrast, instrumentalization refers to adapting a teacher’s teaching method and practices according to the available resources (Brown, 2011). Providing an opportunity for PD, such as mentoring programs, can improve a teacher’s capacity through (a) building quality mentoring of teacher trainees and (b) enabling ground for assessing and deconstructing their teaching practices (Hudson, 2013).

2.8 Instructional accommodation

IA is essential for students with significant disabilities or incapacity, where instructional materials should be modified to suit the student’s level and nature of disability (Hassan et al., 2014). The commonly used examination accommodation for students with special needs includes (a) extended learning time, (b) oral presentation and response, (c) format changes (e.g., large print), (d) relocation to a quiet room, and (e) technology aids (Cawthon, Kaye, Lockhart, & Beretvas, 2012; Lindstrom, 2010). In particular, the effect of the oral examination presentation on the student’s performance varied in different settings. Cawthon et al. (2012) emphasized that the presentation accommodation only benefits the students when a particular examination has high mathematics and language complexity. Thus, there is a need to design the test accommodation according to a student’s specific needs and investigate the linguistic complexity of the examination and other instructional materials. Aside from test accommodation, there are also widely used IAs in mathematics, such as reducing the number of problems, simplifying and enlarging worksheets, minimizing the number of lines on a page, providing more time for completion, peer or cross-age tutoring, raising the number lines, enlarging the number lines, using mnemonic devices, utilizing computational aids (e.g., calculators), using color-coding strategies, and using objects for conceptual understanding (Lindstrom, 2010). However, these IAs are primarily affected by the teacher’s attitude toward inclusion, and programs to promote awareness of inclusion should be a priority for school management (Klehm, 2014).

2.9 Use of technology

The UT in SPED has become a trend for years. This increase in popularity is due to the significant contribution of technology, such as computer-based speech training aid applications, in aiding teachers in dealing with students with disabilities (Cheng & Lai, 2020). Also, the UT has become a current practice for enhancing IA for students with special needs (Lindstrom, 2010). Usability-wise, Eyyam and Yaratan (2014) emphasized that students positively perceive technology in class; thus, continuous use of educational technology would change their attitudes toward its use. Thus, technology must be incorporated into teacher training and supply teachers with the necessary information, aids, and valuable equipment to provide better education, e.g., digital mathematical games (Go et al., 2024). Drijvers (2015) identified three critical factors for successful technology integration in teaching: (a) the design of the technology, the corresponding tasks, and lessons; (b) the role of the teachers in constructing technology-rich activities in the lesson; and (c) a coherent educational context in which the UT can be integrated most naturally. Meanwhile, Kafyulilo et al. (2015) identified factors, such as PD, individual characteristics, and IS, that significantly affect the continuation of technology in a teacher’s pedagogical approach.

2.10 Motivation

A teacher’s work ethic influences teacher motivation, desire to use pedagogical strategies in a school setting, and interest in maintaining order in the classroom (Tambunan, 2018). The scope of teaching motivation includes the motivation to teach and remain in the current profession (Dörnyei & Ushioda, 2011). Motivation has been emphasized in the literature as a significant trait relevant to SPED. Thus, various studies (e.g., Cheung & Kwan, 2021; Yasmeen, Mushtaq, & Murad, 2019) have explored how teachers’ motivation positively impacts students’ learning curves. Accordingly, SPED teachers’ motivation improves if there is an even chance for promotion, increase in salary, allowance, and availability of various facilities (Yasmeen et al., 2019). Furthermore, the component of teacher motivation was expanded by Dörnyei and Ushioda (2011) into four: (a) primary intrinsic motivation or the inherent interest in teaching, (b) social factors regarding the influence of extrinsic conditions, (c) physical change on a serious commitment, and (d) demotivating influences due to negative factors. Teaching enthusiasm also referred to as the intrinsic motivation to teach, has a positive relationship with student outcome, wherein the teacher’s attitude is transmitted directly to the student’s learning behavior (Lazarides, Buchholz, & Rubach, 2018). However, a teacher’s enthusiasm may contribute to biased SF on their learning outcome. There is a tendency for a student to most likely give a positive rating to an expressive and enthusiastic teacher, regardless of their teaching quality (Whitaker, 2011). Nonetheless, high motivation in teaching is significantly related to higher-quality instructional behavior and students’ emotional and cognitive interest in the subject matter (Carmichael, Callingham, & Watt, 2017).

3.1 The Grey System Theory

The “grey” system theory was primarily proposed by Deng (1982) and subsequently by Deng (1989). It is an effective approach for handling uncertain problems with discrete data, which is considered a significant advantage of the grey system over other uncertain environments. Grey numbers and grey systems are discussed in detail as follows. The discussion is presented in such a way that this article is self-contained. Note that the following definitions were lifted from the study by Lin and Liu (2006).

The Converting Fuzzy data into Crisp Scores (CFCS) method introduced by Opricovic and Tzeng (2003) will obtain the equivalent crisp or non-grey matrix of a given grey matrix. Consider the grey matrix X ( ⨂ ) = ( ⨂ x ij ) m × n , where ⨂ x ij ∈ [ ⨂ ¯ x ij , ⨂ ¯ x ij ] . The modified CFCS method is shown in Algorithm 1.

Step 1 (normalization):

Step 2 (calculating the normalized crisp value):

Step 3 (computing the final crisp value):

The equivalent non-grey matrix resulting from the use of Algorithm 1 becomes Z = ( z ij ) m × n .

3.2 The DEMATEL Method

The DEMATEL method is considered a graph theoretic tool for analyzing a structural model or system characterized by elements as vertices and the causal relationships among these elements as edges. Consequent to the direct relationships among elements and their resulting indirect relationships via transitivity, the DEMATEL was designed to categorize all elements into two groups, i.e., net cause and net effect. Such a categorization promotes a better understanding and realization of the system, which may offer solutions to complex problems (Gabus & Fontela, 1972, 1973). Integrating graph theory and linear algebra concepts, the following describes the computational process of the DEMATEL. Note that these steps were lifted from the outline of Mamites et al. (2022).

1. Determine the elements of the system under consideration. This process can be obtained via different approaches, including a literature review on the domain topic, focus group discussion on the practical problem, and expert decisions. Let i = 1 , … , n denote an element of the system.

2. Generate the direct-relation matrix. An expert group of K members performs pairwise comparisons of the causal relationships between n elements. This generates a direct-relation matrix X k = ( x ij k ) n × n where x ij k represents the causal influence of the element i on element j as perceived by the kth member, k = 1,2 , … , K , of the group. An evaluation scale of 0, 1, 2, 3, and 4 is used for the causal influence, representing “no influence,” low influence,” “medium influence,” “high influence,” and “very high influence,” respectively.

3. Aggregate the direct-relation matrices X k , k = 1 , 2 , . . . , K , considering that w k > 0 ∑ k w k = 1 is assigned to the importance of the kth member, as described in equation (15).

   (15) X = ( x ij ) n × n = ∑ k w k x ij k ∑ k x ij k n × n .

4. Normalize the aggregate direct-relation matrix. The normalized direct-relation matrix is calculated using equations (16) and (17).

   (16) G = g − 1 X ,

   (17) g = max max 1 ≤ i ≤ n ∑ j = 1 n x ij , max 1 ≤ j ≤ n ∑ i = 1 n x ij .

5. Calculate the total relation matrix. Once G is obtained, a continuous decrease in the system’s indirect effects along with the powers of G (i.e., G + G 2 + G 3 + …) guarantees convergent solutions to the matrix inversion. The total relation matrix T = ( t ij ) n × n is computed using equation (18), where I is an identity matrix.

   (18) T = G ( I − G ) − 1 .

6. Categorize the elements into the net cause and net effect groups. Compute for D and R using equations (19) and (20), respectively.

   (19) D = ∑ j = 1 n t ij n × 1 = ( t i ) n × 1 ,

   (20) R = ∑ i = 1 n t ij 1 × n = ( t j ) 1 × n .

   The ( D + R T ) vector (i.e., also known as the “prominence” vector) represents the relative importance of each element. Those elements in the ( D − R T ) (i.e., also known as the “relation” vector) having t i − t j > 0 , i = j belong to the net cause group, while those elements with t i − t j < 0 , i = j belong to the net effect group.

7. Create a prominence-relation map. This map illustrates the ( D + R T , D − R T ) mapping of the elements, as shown in Figure 1. The directed relationship of the elements of the prominence-relation map is defined by t ij . However, some of these total relationships are insignificant in theory or practice. To filter out these insignificant relations, a threshold value λ is set. For t ij > λ , a directed edge from element i to element j is drawn in the prominence-relation map. Otherwise, such a directed edge does not exist. The calculation for λ is critical since having a low value implies that most of the relationship is significant, while having a high value suggests that only a few of the relationships are significant.

Figure 1

The prominence-relation map.

4.1 Case Study Background

According to the Institute for Management Development World Competitiveness Ranking, out of 64 countries, the Philippines was ranked fifty-second in 2021 from forty-fifth in 2020. This sharp decline is attributed to the country’s inefficient performance in various areas of concern. Amidst such a dilemma, the Philippine education system faces increasing drop-out and out-of-school rates and shortages of human and teaching material resources (Reyes, Hamid, & Hardy, 2022). At the same time, the system is hampered by a bureaucratic structure that can best be described as dysfunctional and corrupt (Reyes, 2010, 2015). Amplifying such conditions is the evident rise of man-made and natural disasters (e.g., COVID-19), which has shifted the country’s educational system, such as adopting distance learning modalities (Barrot, Llenares, & Del Rosario, 2021).

SPED has always been referred to as a new education paradigm and educational reform goals to make our society more inclusive as part of the global education for all agenda (Allam & Martin, 2021). To assist students with disabilities in making a smooth and successful transition to postsecondary settings, legislation, research, curriculum standards, and teaching techniques have all focused on improving their self-determined behavior traits (Cho, Wehmeyer, & Kingston, 2013). For instance, a legislative measure (i.e., Republic Act No. 7277) was approved in 1992, outlining the provisions for the rehabilitation, self-development, and self-reliance of disabled persons and their integration into mainstream society and for other purposes (National Council of Disability Affairs, 2022). It ensures that persons with disabilities receive quality education and financial assistance. Furthermore, the measure also supports establishing and maintaining a complete, adequate, and integrated system of SPED for the visually impaired, hearing impaired, mentally disabled persons, and other types of exceptional children in all local regions. SPED has recently attracted significant attention to the country’s education sector. For instance, Allam and Martin (2021) quantitatively analyzed the issues and challenges in SPED from the teachers’ perspective. Moreover, Muega (2016) assessed the knowledge and involvement of schoolteachers, school administrators, and parents in educating school learners with special needs in the Philippines. However, the agenda of enhancing the teaching quality of teaching mathematics in SPED remains unexplored in the literature, especially in developing economies such as the Philippines. Thus, this study aimed to advance such an agenda in the Philippine context.

4.2 Application of the Grey-DEMATEL

This section presents the application of the grey-DEMATEL in determining the causal relationships among the success factors of teaching mathematics for SPED students in the Philippines. Shown in Figure 2 is the methodological framework that this study adopted.

Figure 2

The grey-DEMATEL approach.

The computational steps of the grey-DEMATEL approach are discussed as follows:

Step 1. Determine the list of success factors for teaching mathematics in SPED.

The relevant literature is surveyed to determine the list of success factors. Table 1 presents the final list of success factors and their corresponding brief definition and references. The list was then sent to K experts to obtain their judgment about the impact of factor i on factor j .

Step 2. Construct an individual grey direct-relation matrix.

In constructing initial grey direct-relation matrices, a nine-member expert team was invited to this study. The expert team comprises members who have an average teaching experience of 10.7 years in either SPED or mathematics education; two of them have doctorate degrees in SPED, three have Master’s degrees in SPED, one has a Master’s degree in mathematics education, and the rest obtained their undergraduate SPED programs. Four are currently affiliated with universities, teaching college students admitted in SPED or mathematics education programs, while five are in elementary and secondary schools. Due to their rigorous involvement in SPED and mathematics education, the team has the necessary knowledge and expertise to elicit judgments about the factors of teaching mathematics to SPED students.

The influence scores of factor i on factor j ( ∀ i , j ) were elicited from the expert team using a survey questionnaire (Appendix), with a 5-point Likert scale (0: no influence, 1: low influence, 2: medium influence, 3: high influence, 4: very high influence) which forms an individual direct-relation matrix. They are then transformed into grey numbers using a linguistic scale provided in Table 2. For each k th expert ( k = 1,2 , … , K ) , an individual grey direct-relation matrix A k ( ⨂ ) = ( ⨂ a ij k ) n × n where ⨂ a ij k ∈ [ ⨂ ¯ a ij k , ⨂ ¯ a ij k ] is constructed. A sample of the resulting matrix by k th expert is presented in Table 3.

Step 3. Aggregate the individual grey direct-relation matrix.

The individual grey direct-relation matrix A k ( ⨂ ) is aggregated using the formula,

where in w k pertains to the weight of the importance of evaluation elicited by the k th decision-maker such that ∑ k = 1 K w k = 1 . Here, w k = K − 1 . The operations utilized to obtain the aggregate matrix X ( ⨂ ) = ( ⨂ x ij ) n × n are described in Definition 3. The resulting matrix is presented in Table 4.

Step 4. Obtain the normalized grey direct-relation matrix.

The normalized matrix G ( ⨂ ) = ( ⨂ g ij ) n × n is obtained through equations (16) and (17). The resulting matrix G ( ⨂ ) is illustrated in Table 5.

Step 5. Construct the crisp normalized direct-relation matrix.

Algorithm 1 was utilized to transform the grey numbers into crisp values. The resulting crisp normalized direct-relation matrix Z = ( z ij ) n × n is shown in Table 6.

Step 6. Calculate the total relation matrix.

The total relation matrix is constructed using matrix Z through equation (18). The resultant matrix is presented in Table 7.

Step 7. Compute the prominence and relation vectors.

In this step, the dispatched ( D ) and the received ( R ) of each factor are obtained using equations (19) and (20), respectively. Then, the ( D + R T ) and ( D − R T ) vectors are determined. If ( D − R T ) < 0 , the factor is considered a net effect, and if ( D − R T ) > 0 , the factor is considered a net cause. Aside from the category, the ranking of the factors is also defined using the ( D + R T ) .

Step 8. Construct the prominence-relation diagram and determine the significant causal relationships.

The prominence-relation diagram is constructed using ( D + R T , D − R T ) as coordinates of each factor. Then, the significant causal relationships among the success factors of teaching mathematics in SPED were identified. Here, λ is defined as the 75 th percentile, chosen so that the resulting causal network portrays those more compelling relationships among factors. Hence, if the values of the total relation matrix t ij are at least equal to λ , then the causal relationship is significant. The prominence-relation map, including the identified causal relationships, is shown in Figure 3, with 25 relationships.

Figure 3

Prominence-relation map of the factors of teaching mathematics for SPED students.

4.3 Sensitivity Analysis

The sensitivity analysis of the grey-DEMATEL for evaluating the CSFs of teaching mathematics to SPED students is presented in this section. The sensitivity analysis performed in this section would determine if a different set of linguistic scales would change the category and prominence ranking of the factors. Shown in Table 8 are the different linguistic scales used in determining the robustness of the method.

The result of the analysis is presented as compared to the baseline result obtained from this study. As shown in Figure 4, the category (i.e., net cause, net effect) only has minimal changes in the three scenarios. Notably, the values of the scenarios follow the same trend as the baseline. Moreover, as featured in Figure 5, the prominence ranking of the three scenarios has little to no changes compared to the baseline values. The results reveal that in all cases, the prominence and relation values remain unchanged with the introduction of different linguistic scales. The sensitivity analysis revealed robust and valid results, implying that the study’s findings are reliable for decision-making.

Figure 4

Comparative results on the category of factors.

Figure 5

Comparative results on the ranking of factors.