On a very basic level, student engagement is the measurement of how many students are participating actively in a lesson. Student engagement is one the most important parts of my job as a teacher. A teachers ability to reach every student in the room and teach in a way that includes all and resonates with a diverse group of students is essential to succeeding as an educator. If a student is not engaged in a lesson, then they are not learning.
On a deeper level, I think engagement is the result not the process of the educator. I think that great student engagement comes from investment in, rapport with, and respect for students. The greatest teachers in my past were those who took and interest in me and invested in who I was as a person, not a student. A teacher should engage their students by investing in who they are as people. No matter how talented or smart someone may be, their words will never be heard if they are cruel and unkind. Building a rapport with students that is grounded in reliability, kindness, and understanding is essential to have students engaged in your lessons. Finally, respect is the corner stone of the musical classroom. However, I am not talking about students respecting the teacher. I am talking about the teacher respecting the students. My father would always tell me "You cannot earn respect, you can only give it." These are words I have lived my life by. A teacher can not hope to have the students respect in the classroom without first respecting the students.