In my reflection on this reflection, I came up with four main categories from my experience that describe why students have been interested in or motivated to study and learn mathematics.
Patterns and beauty inherent in mathematical structures
Some students are intrigued by looking for, identifying, and explaining patterns; others enjoy the beauty inherent in visual representations of mathematical objects and relationships. These students appreciate a teacher who encourages and rewards their curiosity, but overall, require the least amount of effort on the teacher's part to motivate and support since they're often speaking the same language as the teacher already.
Applications between mathematics and the real world
Other students I have taught were less interested in math in and of itself, but did find the idea of math as a tool to understand, explain, and predict the real world motivating. These were often students with an existing interest in science or social science who saw the usefulness of math in their respective fields of interest. Interesting projects were obvious choices in hooking and motivating these students, as well as a greater emphasis on practice and application than on derivation or justification.
Being a good studentThis third category of student is one that is invested in an image of themselves as a good student. They care about doing well and meeting their goals and are motivated by seeing their progress, exerting effort and seeing it pay off, as well as specific feedback on how to improve and clear objectives for the course.
RelationshipsThese students seem to be predominantly motivated by positive interactions with others, whether that's the teacher or their peers. Classroom structures that increase conversations and collaboration between students and that make students feel known and connected to others have been helpful in motivating this group in my experience, as well as putting more of an emphasis on my relationship with them.
Obviously, most students are some mix of these categories, but for many in my experience, one is more dominant. I think that a classroom that tries to balance between these different student needs will likely result in broader student success than one that caters to only one type. I would love pushback on my preliminary and perhaps too simplified analysis. Are there any categories you see being useful for thinking about student motivation? What other tools and strategies have you used to help students foster their curiosity and interest about math and motivation to exert effort towards the class?