"Under the guidance of the teacher" 用英语可以说成 "under the leadership of the teacher" 或 "under the direction of the teacher"。
"Leadership" 来源于英语单词 "lead"，意思是带领、领导。"Direction" 来源于英语单词 "direct"，意思是指导、指引。
例句：
Under the leadership of the teacher, we were able to complete the project successfully. (在老师的带领下，我们成功完成了项目。)
Under the direction of the teacher, we were able to improve our Engpsh skills. (在老师的指导下，我们成功提高了英语水平。)