One of my favourite quotes about mathematics: “Mathematical notation no more is mathematics than musical notation is music. A page of sheet music represents a piece of music, but the notation and the music are not the same; the music itself happens when the notes on the page are sung or performed on a musical instrument. It is in its performance that the music comes alive; it exists not on the page but in our minds. The same is true for mathematics.” — Keith Delvin
Are your students really DOING mathematics, or just working with notations on a page?