For reference, I am in mathematics. I will be on the job market again, and I need to write up a new research statement. I have interests in the area of pure mathematics that I did my thesis research on, but I also have interests in research in math education and also research in mathematics and art.
Does it reflect bad on me if I express my wide range of research interests in my research statement?
I am mostly interested in teaching positions, possible with some research, like a liberal arts college.
I am afraid it looks weird because these are quite disjoint from each other, although I have ongoing projects in each area, I am afraid may look like I'm scatterbrained.
It depends what they are looking for. Some employers will place a premium on flexibility, in which case your variety of interests will be a positive. It will help you to build connections with other departments, potentially increasing opportunities for service teaching, and hence increasing funding to the department. It will enable you to go with the flow, to seek research funding in whatever area has the best prospects at the moment, and collaborate on a variety of projects inside and outside your department. It will also help you to teach from several different angles.
Other employers are looking to build on an existing strength (for instance, a strong research stream in a particular area or good relationships with a particular external stakeholder) or fill a particular gap in their existing strengths. In this case, a variety of interests will be a negative, because you might shift focus away from the research interest for which they are really hiring you. Usually, job ads will mention whether there is a particular specialty that they are really after, but it also doesn't hurt to ask around to find out if you are not sure.