The Vietnamese word "nảy sinh" means "to develop" or "to appear." It is often used to describe the emergence of ideas, feelings, situations, or problems.
The phrase you provided, "phải giáo dục trẻ em, đừng để nảy sinh ra tính tự kiêu," translates to "one must educate children and prevent self-conceit from developing in them." This illustrates how "nảy sinh" can refer to negative traits or tendencies that should be avoided through proper education and guidance.