I have included not only information about what constitutes a legitimate dance teacher and what a choreographer is, but also information about what a dance teacher is not, and the predators in in dance instruction who do a world of harm to the profession and art of dance.
It is true that dance teachers often are ex-professional dancers, and require teacher’s training and pedagogy, so they can take that experience into a studio and translate it through a curriculum to train beginners through advanced students, possibly to professional. This is what is expected of any teacher of any skillset. Indeed, this is why academic school teachers must be certified to teach in their fields before being released to a classroom of students.
However, first, most schools have a recreational bent, and are not focused upon creating professionals. Thus, they do not train, coach and rehearse the students the 35 or so hours per week it takes to build a finished entry level professional dancer. This is fine, as professional schools usually require audition and examinations for entry that refines their enrollment to those with professional desires, talent and ability.
But, even in recreational schools, the teachers must still have the experience and training to work with all kinds of students. Fact: Because there is no legal certification for teachers, or an accurate test of ability, 90 percent of the teachers in the US do not come close to the required experience or education it takes to teach anyone! (if you have seen the TV show "DanceMoms" you see this incompetence in action in every episode.)
So, when it comes to looking for a dance school, caveat emptor! The first factor a parent can use to rule out a school is one that specializes in taking their children to amateur talent "conventions" and pageant/competitions. The vast majority of these very commercial competitions are all about how much money they can glean from the parents and schools, who often in turn raise the cost to take a percentage of the fee, costume fees, teacher fees, choreography fees, etc.
The studios that engage in these unethical practices are usually and rightly referred to as "studios", because they are not really schools, even though they may advertise themselves this way. (Note, that there are many that do not engage in the profit taking, but still participate in competitions; still, 'tis best to wary.)
These are what are known by consumer watchdog groups as "predatory businesses". They are legion in the dance world, and far outnumber legitimate training and recreational institutions. Often, they have names like "StarDance," "Showcase," "Footloose" or the like. Do your research, because these so-called "schools" (studios) spend no time teaching (mostly because they don't know how to teach), and a huge amount of time setting "routines", ("Routine" is an apropos statement about what they actually do. "Boorish" would be an even better term).
Caveats: Make sure the school does not combine dance genres into one class, for example, tap, jazz, acrobatics and ballet in one class. Make sure there are no multiple layers of fees. Make sure they have separate classes for training the dancers, from rehearsals for performances, or these predatory competition pageants. Check the teachers’ and directors’ credentials beyond their website! If they have no real professional experience combined with university degrees or extensive training with mentors, in pedagogy, management and the arts, beware. They should have studies in dance methodologies, dance history, human movement potential, anatomy and kinesiology, theater arts and craft, aesthetics, classroom practicum etc.
If the school concentrates on teaching children to be artists for performance, then it is worth further examination of the teacher's credentials. Many of these are not great schools, as their teachers barely have enough background to justify their career. But at least they are working towards a sense of excellence.
Choreographers are not necessarily dance teachers. Indeed, whilst choreographing or "setting" dances, they are in no way instructors. In the sense that they resemble teachers, when choreographing, they are staging dances. As such, they are not directly "teachers.” Staging is a form of teaching, because they are composing steps, staging and movement patterns for the dancers to memorize, to make a cohesive work of art. But, this is specifically the transmission of "kinetic contextualization" so the dancers learn their parts, similar to an actor learning a part by memorizing their lines, or a singer memorizing a song or aria. This is much different than teaching as instruction, which is what a "dance teacher" does, or should do.