Electronic Theses and Dissertation

Pemodelan dalam matematika adalah suatu usaha untuk menyederhanakan permasalahan nyata dari variabel yang memiliki keterhubungan. Salah satu teknik analisis data yang menggunakan pemodelan adalah Partial Least Square-Structural Equation Modeling (PLS-SEM). PLS-SEM disebut sebagai pemodelan kausal prediktif yang menjelaskan hubungan antar variabel dalam bentuk diagram jalur (path model). Jalur dalam pemodelan PLS-SEM menghubungkan variabel dengan didasarkan pada teori yang ada.
Salah satu masalah yang sering diperdebatkan dalam analisis pemodelan dengan PLS-SEM adalah ukuran sampel. Perdebatan ini mendorong penulis melakukan analisis berbagai aspek yang seharusnya menjadi mempertimbangkan dalam menggunakan suatu ukuran sampel tertentu, dan juga metode penentuan sampel yang digunakan dalam PLS-SEM.
Terdapat tiga metode penentuan ukuran sampel yang disarankan dalam PLS-SEM, yaitu ten times rule, inverse square root dan gamma exponential. Jika ten times rule tidak menggunakan perhitungan matematika dalam penentuan ukuran sampel, maka inverse square root dan gamma exponential menggunakan perhitungan matematika dalam penentuan ukuran sampel. Penentuan ukuran sampel dengan dua metode tersebut menggunakan pertimbangan minimal path coefficient yang diharapkan pada sebuah model jalur. Path coefficient pada model PLS-SEM mewakili efek yang diberikan satu variabel eksogen terhadap variabel endogen.
Ukuran sampel yang dihasilkan melalui dua metode tersebut, sekalipun menggunakan nilai path coefficient yang sama, namun menghasilkan ukuran sampel yang berbeda. Asumsinya, semakin kecil path coefficient yang diharapkan dapat diamati pada model, maka semakin besar ukuran sampel yang dibutuhkan. Dengan menggunakan path coefficient 0,2, 0,3, dan 0,4 penulis menemukan ukuran sampel 39, 69, dan 155 dengan menggunakan metode inverse square root, dan ukuran sampel 26, 56, dan 141 dengan menggunakan metode gamma exponential.
Teknik analisis data dengan menggunakan pemodelan dalam PLS-SEM menurut ahli efektif digunakan untuk ukuran sampel yang kecil. Namun demikian, peneliti tetap harus memperhatikan seberapa kecil ukuran sampel yang sesuai dan yang dapat menggambarkan hubungan antar variabel pada model dengan lebih baik. Menganalisis hubungan antar variabel pada model dalam PLS-SEM meliputi koefisien determinasi (R2), validitas indikator, validitas model, effect size, croos loading, dan nilai signifikansi dari path coefficient yang menggambarkan efek suatu variable terhadap variabel lainnya.
Terdapat dua variabel yang digambarkan dalam pemodelan PLS-SEM, yaitu variabel eksogen yang mempengaruhi variabel lainnya dan variabel endogen yang dipengaruhi oleh variabel eksogen. Untuk penelitian ini penulis menggunakan aspek mathematics teacher beliefs, yaitu beliefs about nature of mathematics, beliefs about teaching mathematics, dan beliefs about assessment in learning mathematics sebagai variabel eksogen dan karakteristik guru sebagai variabel endogen.
Berdasarkan hasil analisis yang dilakukan didapat bahwa ukuran sampel N=141 yang ditentukan berdasarkan metode gamma exponential memberikan nilai hubungan antar variable yang lebih baik dibanding ukuran sampel lainnya (26, 39, 56, 69, dan 155). Berdasarkan analisis ini menunjukan bahwa ukuran sampel yang lebih besar belum tentu memberikan gambaran mengenai hubungan antar variabel pada PLS-SEM yang lebih baik.
Mathematics teacher beliefs dianggap dapat memberikan pengaruh terhadap karakteristik guru, terutama karakteristik profesional guru matematika terkait profesinya sebagai guru yang melaksanakan pembelajaran di kelas. Dikarenakan belum terdapat path model yang menggambarkan hubungan mathematics teacher beliefs terhadap karakteristik guru yang dikembangkan saat ini, penulis membandingkan 3 path model dengan hubungan jalur yang berbeda. Agar lebih valid, penulis menghilangkan indikator dengan nilai loading < 0,5. Hasil analisis menunjukkan bahwa semakin kompleks jalur yang menghubungkan antara variabel, maka semakin baik path model tersebut digunakan karena dapat memberikan informasi yang beragam dengan berbagai alternatif sudut pandang tentang hubungan mathematics teacher beliefs terhadap karakteristik guru. Ketiga path model dapat digunakan sesuai dengan tujuan dan data yang digunakan, namun penulis menyarankan bahwa path model 2 lebih dipertimbangkan digunakan lebih lanjut dibandingkan dua model lainnya.
Pada path model ini aspek beliefs about teaching mathematics dan beliefs about assessment in learning mathematics diberikan peran sebagai variabel mediasi antara beliefs about nature of mathematics dengan karakteristik guru. Hasil analisis menunjukkan bahwa pengaruh tidak langsung beliefs about nature of mathematics yang lebih besar terhadap karakteristik guru dibandingkan pengaruh langsungnya. Beliefs about teaching mathematics dan beliefs about assessment in learning mathematics lebih potensial menjadi variabel mediasi secara bersama terhadap efek beliefs about nature of mathematics pada karakteristik guru dari pada menjadi variabel mediasi secara terpisah. Hal inilah yang menjadi pertimbangan bahwa path model 2 lebih potensial digunakan, karena pada model ini, jalur yang digambarkan bukan hanya menggambarkan efek langsung namun juga menunjukkan efek tidak langsung antar variabel.
Berdasarkan hasil penelitian ini, penulis mengharapkan lembaga pendidikan guru matematika ataupun pusat pengembangan kompetensi guru matematika yang berfokus pada pengembangan karakter professional guru, dapat mempertimbangkan aspek mathematics teacher beliefs dalam pengembangan kurikulum dan pendekatan pembelajaran yang direncanakan. Beliefs about nature of mathematics sebagai pengetahuan guru tentang matematika perlu didampingi peningkatan keyakinannya terhadap pembelajaran dan asesmen yang dilakukan di kelas. Hal ini diharapkan dapat berjalan bersamaan dalam pendidikan calon guru atau pengembangan kompetensi guru matematika di masa depan.

Modeling in mathematics seeks to simplify real-world problems by analyzing variables that exhibit relationships. One prominent data analysis technique that employs modeling is Partial Least Squares Structural Equation Modeling (PLS-SEM). PLS-SEM is recognized as a predictive causal modeling approach that elucidates the relationships between variables in the form of a path model. In PLS-SEM, the paths connecting variables are based on established theories. A frequently debated issue in PLS-SEM analysis is the determination of sample size. This debate has prompted the author to explore various factors to consider when selecting an appropriate sample size and the methods for sample determination employed in PLS-SEM. Three methods for establishing sample size are commonly suggested in PLS-SEM: the ten times rule, the inverse square root method, and the gamma exponential method. While the ten times rule does not rely on mathematical calculations for sample size determination, both the inverse square root and gamma exponential methods do incorporate such calculations. The choice of sample size using these two methods takes into account the minimum expected path coefficient in a given path model. In PLS-SEM, the path coefficient represents the effect an exogenous variable has on an endogenous variable. The sample sizes generated by the two methods, despite using the same path coefficient value, result in differing outcomes. The underlying assumption is that a smaller expected path coefficient in the model necessitates a larger sample size. For path coefficients of 0.2, 0.3, and 0.4, the author identified sample sizes of 39, 69, and 155 using the inverse square root method, while the gamma exponential method yielded sample sizes of 26, 56, and 141. Experts suggest that data analysis techniques employing modeling in PLS-SEM are effective for small sample sizes. However, researchers must still consider the minimum appropriate sample size that adequately captures the relationships between the variables in the model. Analyzing these relationships in PLS-SEM involves examining the coefficient of determination (R²), indicator validity, model validity, effect size, cross-loading, and the significance value of the path coefficients that illustrate the influence of one variable on another. In PLS-SEM modeling, there are two types of variables: exogenous variables, which influence other variables, and endogenous variables, which are influenced by exogenous variables. In this study, the author focused on aspects of mathematics teachers' beliefs—specifically, their beliefs about the nature of mathematics, their beliefs about teaching mathematics, and their beliefs about assessment in learning mathematics—as exogenous variables, while teacher characteristics served as the endogenous variables. The analysis revealed that the sample size of N = 141, determined using the gamma exponential method, provided a more favorable relationship between the variables compared to the other sample sizes (26, 39, 56, 69, and 155). This analysis indicates that a larger sample size does not necessarily yield a clearer understanding of the relationships among variables in PLS-SEM. Data analysis techniques utilizing modeling in Partial Least Squares Structural Equation Modeling (PLS-SEM) are recognized by experts as effective for small sample sizes. Nevertheless, researchers must be mindful of the appropriate sample size and its ability to accurately capture the relationships among the variables within the model. When analyzing these relationships in PLS-SEM, important aspects to consider include the coefficient of determination (R²), indicator validity, model validity, effect size, cross-loading, and the significance of the path coefficients, which illustrate the impact of one variable on others. In PLS-SEM modeling, there are two types of variables: exogenous variables, which influence other variables, and endogenous variables, which are affected by exogenous variables. In this study, the author examined aspects of mathematics teachers' beliefs—specifically beliefs regarding the nature of mathematics, beliefs about teaching mathematics, and beliefs about assessment in mathematics learning—as exogenous variables, while teacher characteristics were treated as endogenous variables. The analysis revealed that a sample size of N = 141, determined through the gamma exponential method, yielded a superior relationship value between variables when compared to other sample sizes of 26, 39, 56, 69, and 155. This finding indicates that a larger sample size does not necessarily provide a clearer understanding of the relationship between variables within the context of Partial Least Squares Structural Equation Modeling (PLS-SEM). Mathematics teacher beliefs are recognized as influential factors impacting teacher characteristics, particularly the professional attributes of mathematics educators in the classroom setting. Due to the absence of an established path model that delineates the relationship between mathematics teacher beliefs and teacher characteristics, the author compared three distinct path models featuring varying path relationships. To enhance the validity of the analysis, indicators with loading values less than 0.5 were excluded. The results demonstrated that increased complexity in the paths connecting the variables correlates with improved effectiveness of the path model, as it provides multifaceted information and various alternative perspectives regarding the relationship between mathematics teacher beliefs and teacher characteristics. While all three path models are applicable based on the specific objectives and data employed, the author advocates for the consideration of path model 2 for further application over the other two models. In this particular model, the dimensions of beliefs regarding mathematics instruction and assessment are positioned as mediating variables between beliefs about the nature of mathematics and teacher characteristics. The analysis indicates that the indirect influence of beliefs concerning the nature of mathematics on teacher characteristics is more pronounced than its direct influence. Furthermore, beliefs about teaching mathematics and beliefs about assessment serve as more effective joint mediating variables in the relationship between beliefs about the nature of mathematics and teacher characteristics than when treated as separate mediating variables. This substantiates the premise that path model 2 holds greater potential for application, as it illustrates not only the direct effects but also the indirect effects among the variables. Based on the findings of this study, the author anticipates that mathematics teacher education institutions and centers focused on developing teacher competencies should prioritize the enhancement of professional teacher character through the inclusion of mathematics teachers' beliefs in the design of curricula and learning strategies. It is essential that teachers' understanding of the nature of mathematics is complemented by a strengthening of their beliefs regarding classroom learning and assessment practices. This dual approach is expected to be implemented concurrently in the education of prospective teachers and in the ongoing development of mathematics teacher competencies in the future.