## 1 Introduction
It is important to mention that we considered design factors in TPDs for technology integration and teachers' creativity. These variables, like the 'effective TPD' parameters, interact horizontally in our training. In the last phase of the WOA, which is called exploration, we enhanced our design with a different cooperative strategy to examine further effectiveness in our training. The bibliography revealed that the Team-Based Learning (TBL) strategy would be suitable for our purpose. This active learning strategy boosts subject knowledge, comprehension, engagement, critical thinking, and participation among group members. TBL offered coaching and support, which are components of the effectiveness of TPDs. The training was designed with these components, as the WOA methodology and the literature review have indicated to us during the design process. To test our variables, we conducted an MCMC sampling technique. Combining these methods to find optimal training solutions in educational research remains a novel approach. The research questions derived from the above are the following:  
RQ1: Is there a combination of methods, resources, content, assessment, and duration that is optimal when designing a TPD for creativity in technology-enhanced learning design? RQ2: Can WOA be used as a method to obtain optimal solutions when designing training in creative learning design? RQ3: Can MCMC help us validate this training in creative learning design in a TPD, giving the expected accuracy to our proposed results?The article is as follows: In the second section, we present the related literature for the WOA and MCMC optimization algorithm and their applications in educational research. In the third section, we present our methodology. In the fourth chapter, we analyze the conceptualization of WOA in our training. In the fifth section, we present the mathematical formulation and the algorithmic precision as combined with the MCMC application. In the sixth section, we present the results. Lastly, in the final section, we show the conclusions and future remarks of this article. Whale Optimization Algorithm, Markov Chain Monte Carlo, and Their Applications in Educational Research  
The WOA is a metaheuristic algorithm that draws inspiration from the bubble-net feeding technique of humpback whales, in which they generate spirals of bubbles to ensnare prey before striking. In 2016, [PERSON] and [PERSON] introduced whale swarm optimization methods. The key concepts of WOA [14] are:  
* Encircling Prey: The optimal solution is viewed as the prey. Other whales modify their locations concerning this optimal solution through mathematical equations. * Bubble-net Attaching (Exploitation): Two mathematical models have been suggested to simulate the whale's behavior while hunting their prey: the shrinking encircling method and spiral position updating. In this phase, which is called the exploitation phase, whales enhance their search for the optimal solution by either reducing their movement range or adhering to a spiral pattern. A probability factor (50%) governs which approach is utilized. * Shrinking Encircling: The whales gradually converge towards the best solution. * Spiral Updating: Whales move in a spiral formation toward the optimal solution. * Search for Prey (Exploration Phase): Whales randomly choose a target and adjust their movements toward or away from it. This method guarantees a global search for the most effective combination. WOA has been implemented in education, especially in machine learning, but also in other educational research, involving student performance. For example, the article named \"An Enhanced Evolutionary Student Performance Prediction Model Using Whale Optimization Algorithm Boosted with Sine-Cosine Mechanism\" presents a refined approach to forecasting student success by augmenting the Whale Optimization Algorithm (WOA). This improvement combines the Sine Cosine Algorithm (SCA) and Logistic Chaotic Map (LCM) to tackle WOA's shortcomings in both the exploration and exploitation stages [15].