When I was a student, a typical school day consisted of my teacher talking to the class for an extended time of direct instruction, followed by him/her telling the whole class what one required assignment we all had to complete to demonstrate our understanding. We each sat in identical chairs at identical desks that the teacher had assigned to us, and the desired outcome was that we all completed our worksheets (it was almost always a worksheet) in the identical "correct" way. We were rarely allowed to choose anything during the school day, because the teacher was the expert and we were supposed to be passive sponges, absorbing knowledge.
Unfortunately I know that some teachers/principals/schools still operate this way today.
As a teacher, I see the value in allowing and outright encouraging choice in my students. They choose whether they work at a desk, at a large table, on a rug, in the loft bed, hugging a pillow, wrapped in a blanket, or in another place/way. My students choose whether they build a model of a math concept, solve problems on white boards, play a game to practice the skill, draw a picture to demonstrate their methods, provide examples/non-examples of a term, or make up their own, completely novel solution. My students choose to make board games, construct dioramas, write sequels, write and perform dramas, compose poetry or rap, or draw pictures to demonstrate their comprehension of books.
It makes me sad to think that encouraging choice would even have to be suggested for a way to create a culture of innovation. If our ultimate goal as teachers and other educators is to create an environment where our students can grow to be ready for the world outside of our classrooms, shouldn't choice be a part of every day, every class, and every lesson? I humbly suggest yes.
(For this idea suggestion I choose the "other" tab, because choice can apply to all of the topics)