## 3 Methodology
### Data Analysis and Model Validation  
The fuzzy system is generated to address the first research problem using the MATLAB R2021b Fuzzy Logic Toolbox. To predict teachers' MPSB scores, artificial neural network (ANN) and adaptive neuro-fuzzy inference system (ANFIS) models are established. The implementation of the fuzzy inference systems in this study used triangular membership functions due to their simplicity and computational efficiency. Each input variable (e.g., gender, age, level of education, school level, teaching experience, and creative thinking disposition) was associated with a set of triangular membership functions. These functions were defined based on empirical observations of the input ranges and were adjusted iteratively during the fuzzy system development. Additionally, alternative membership function types such as trapezoidal and Gaussian functions were considered during the initial exploratory phase, but the triangular functions provided the best balance of simplicity and model accuracy for this dataset. The fuzzy inference system (FIS) was constructed using MATLAB's Fuzzy Logic Toolbox. Input variables and membership functions were defined within the toolbox, while rules were configured manually and tested iteratively. A hybrid optimization approach combining back-propagation and least squares estimation was used for training ANFIS models. This approach adjusted membership function parameters effectively to minimize error during training. ANN models were implemented using a feedforward architecture with one hidden layer. The number of neurons in the hidden layer varied from 1 to 10 to identify the optimal configuration that minimized RMSE and maximized performance. Each neuron used a logarithmic sigmoid activation function to introduce non-linearity, while the output layer employed a linear activation function for regression. Back-propagation with the batch gradient descent method was used to update the weights and biases iteratively. This algorithm minimized the mean squared error between predicted and real scores. Input data were normalized to the closed interval (0,1) using min-max normalization before training to ensure consistency and improve convergence speed. During training, the model's RMSE and coefficient of determination (\(R^{2}\)) were monitored. The model achieving the lowestvalidation RMSE while avoiding overfitting was selected as the final architecture. A subset of the data not used during training was utilized for validation, ensuring the network's generalization ability was evaluated independently. Testing was conducted to confirm predictive accuracy on unseen data. To evaluate the models, several criteria were employed to ensure robustness and validity:  
1. Spearman-Brown Correlation Coefficients: These were used to measure the relationship between real scores (calculated from the Mathematical Problem-Solving Beliefs Scale) and artificial scores predicted by the ANFIS and ANN models. Correlation coefficients were interpreted as follows: 0.00-0.30 (low), 0.30-0.70 (moderate), and 0.70-1.00 (high). 2. Training and Testing Data Partitioning: The dataset was split into training (70%), validation (15%), and testing (15%) subsets to rigorously evaluate the performance of the models. This partitioning allowed us to analyze overfitting and generalization capabilities effectively. 3. Membership Function Validation: For fuzzy models, defuzzification techniques (such as weighted average methods) were employed to ensure precise mapping between inputs and outputs. Predicted results were cross-checked with real values to confirm alignment within acceptable error ranges. In the second research problem, it was examined separately whether there was a relationship between teachers' real MPSB scores and artificial MPSB scores created with ANFIS and ANN approaches.