## 1 Introduction
Another example is a hybrid application of WOA in Teacher Professional Development (TPD). The article 'Teaching learning-based whale optimization algorithm for multi-layer perceptron neural network training' presents a valid way to optimize the multi-layer perceptron neural network [16]. WOA could serve to improve training programs by identifying the most effective training modules for educators. It potentially aids in TPD optimization, as methods, resources, assessment, time, context, and processes must be fluidly adjusted for teacher development. Computational modelling focuses on building progressively better models of phenomena rather than collecting individual results. In our case, it will be a suitable enhancement in educational research to utilize them for a dynamic output like the creation of a TPD in technology and creativity [17]. Algorithms help us graph this iterative process in an 'easy-to-use' way, providing at the same time more accurate starting points because of their flexibility to evolve during the process. The WOA has its mathematical models and applications. In this article, we will present a novel application of WOA combined with the MCMC. Monte Carlo methods comprise a set of computational techniques that depend on the outcomes of numerous random experiments to determine an unknown value. MCMC is a collection of sampling methods. MCMC was initially developed in 1953 and experienced significant growth in the 1990s. The origins of MCMC can be traced back to chemical physics, while its advancement was supported by applied mathematics (statistics), leading to its application in various fields, including social sciences. By the 20 th century, MCMC sampling techniques had become widely recognized in academia. MCMC refers to algorithms that implement Monte Carlo sampling using a Markov chain. Conventional Monte Carlo techniques sample independently. In comparison, MCMC possesses a Markovian property: the following selection relies on the prior one in the sampling procedure [18]. As discussed, MCMC entered social studies in 1990 as a quantitative analysis method in research. In education, we have various practical applications of MCMC. For example, an application in education is about student performance prediction. In the paper, 'Mathematical Model for Early Prediction of Learning Success' [19], there is an attempt to address the challenges of overseeing educational success in real-time, regardless of the discipline, and with great accuracy. The model utilizes information from Learning Management Systems (LMSs), which monitor students' engagement, advancement, and performance instantly. MCMC helped define these valuable elements from the data and then used them. Another important application of MCMC in education is when designing assessments. For example, in the recent study, 'Markov Chain Monte Carlo Methods for Dynamic Student Assessment' [20], we have the utilization of the MCMC in creating the best possible assessment in real-time. Thanks to MCMC techniques, the model can estimate complex probability distributions, allowing for adaptive assessment and monitoring of each student's progress. At the same time, it provides quantification of uncertainty in estimates, making the assessment more accurate and personalized. Lastly, we have recent examples of research in teacher decision-making and practices that utilize the MCMC methodology. The paper 'The Application of Markov Chain in Middle School Mathematics Teaching Evaluation' presents how the MCMC can analyze teaching effects and guide teachers to improve their teaching methods in mathematics. By creating an initial state vector and a transition probability matrix, the research assesses variations in student performance among various classes. The results show that this method offers an objective way to evaluate teaching effectiveness and can assist educators in improving their instructional strategies. In particular, the research revealed that this is a good quantitative method to assess teacher quality [21].