Large language models demonstrate remarkable proficiency in various tasks across multiple languages. However, their potential in code-switching remains underexplored, particularly in cultural and educational contexts. Code-switching or translanguaging plays a crucial role in bilingual education, facilitating comprehension and engagement among students with varied linguistic proficiencies. In this work, we present a pedagogy-inspired framework that introduces traditional classroom practices of code-switching to intelligent tutoring systems. Specifically, we develop fine-grained instructional strategies tailored to multilingual and educational needs. We conduct experiments involving both LLM-based evaluation and expert analysis to assess the effectiveness of translanguaging in tutoring dialogues. Our experimental results indicate that strategic code-switching can significantly enhance the learning experience. This work not only advances dialogic tutors in language learning, but also extends LLMs to better accommodate multilingual interaction.