One of the things students often want, and parents almost always want for their children, is for piano to become developed as a creative outlet. What "creative outlet" means isn't always well-defined, but I like how piano pedagogue Forrest Kinney split it into the four arts of improvisation, arranging, interpretation, and composition. I believe that most classical piano teachers view their students' development of a creativity as highly desirable, and yet I've observed that it nevertheless gets sidelined in many teaching approaches, especially when it comes to improvisation, arranging, and composition. I see a couple primary reasons for this:
Most piano teachers have little time with their students compared to subjects of comparable complexity, such as math, reading, or swim team. 30 minutes once a week is not uncommon. Short lesson times mean sometimes just teaching a core skill is all there's time for, and maybe the time runs out before the imagination is invited to mingle with the new skill. Teachers also sometimes have a sense that if they focus on skills just related to creativity, the students aren't progressing along the curriculum
Our culture lacks a pedagogy of creativity. Teachers struggle with benchmarking what it means to improve. And it can be hard to find resources. Piano teaching conferences have for many years had sections on "incorporating creativity"--like what you can try first if you're doing nothing--but not as much information on what to do next, and after that, let alone how to structure a years-long sequential curriculum. Pedagogy textbooks and student method books offer surface-level activities along the same lines, only even fewer of them than in the conference sessions. Exams and competitions don't usually make space for student compositions or improvisations, and most adjudicators wouldn't be equipped to handle them if they did. College auditions for classical music programmers also aren't structured to benchmark creativity--they are much more interested in measuring technical skill and ability to convey musical structure.
Though these challenges are formidable, I think there's things we can do as teachers to work through them so our students can be expressive at their instruments. Here's stuff I've found useful:
It's better to have longer or more frequent lessons. I've for years chosen not to teach anyone less than 45 minutes per week. Sometimes that means 30 or 45 minutes twice a week, and sometimes it means 45, 60, or 90 minutes once a week. Having sufficient teacher time is a key factor in making piano lessons really work out for any student.
I've found it wildly helpful to read books and attend workshops on pedagogies of creativity. Edwin Gordon was a researcher with a sequenced approach to rhythmic and tonal solfege that I've found helpful and use with students. Forrest Kinney also has excellent books on the subject, including books for direct use with students. I've been getting more and more into various jazz resources lately as well, like the work of Jeremy Siskind. David Berkman's Jazz Harmony Book has transformed how I teach harmonization, even when we aren't talking about jazz. There's also great resources from early music, since various times and places of the past had robust pedagogies of creativity. These early music resources are called partimenti, and Job IJzerman has a great book on getting started with them. In the Netherlands, many universities are now teaching partimenti based on the Izerman textbook in place of their former harmony 101 courses.
I generally avoid competitions and exams, especially for young students. I instead host three recitals a year, and at each one, include an open mic improv duet section at the end, where students get a chance to musically interact with each other on stage. This helps send the message that improv is a core skill. Many students are also performing their own compositions and arrangements at the recital, in addition to works by other composers.
At a young age, I start students off with lots of transposition activities so they quickly get intimately acquainted with all 12 keys. They also play by ear more and more complicated tunes. As they move along, they learn to harmonize tunes, and to do so with more and more complexity. For holiday time, I like having students (who wish to participate) learn their favorite holiday songs and besides just learning new ones, also arrange the previous ones in a more complicated way every year, and in more keys, so that they can feel the progression, and also have important family heritage repertoire at their fingertips.
Improv begins early with the idea of a conversation. We play back and forth on the piano with the value that whatever each person does must be in response to what their duet partner is doing--even if we just start simple by playing one key back and forth. As they develop improv skills, one day, they inevitably have an improvised idea they like enough to repeat in different ways, and that's how composition begins.
Being expressive is a core part of being human, and it's wonderful we get to have these elevated expressive experiences at the piano.