Your teacher introduces the concept of Transcendentalism, a literary movement that believes truth exists beyond reason and experience. Then she asks you to determine the author's purpose for this quote by Henry David Thoreau:

"If a man does not keep pace with his companions, perhaps it is because he hears a different drummer. Let him step to the music which he hears, however measured or far away."

The purpose of David Thoreau was to speak for each one to follow his own intuition and not to be led by influences.

Explanation:

Transcendentalism is the name of the group of new ideas in literature, religion, culture, and philosophy that preaches the existence of an ideal spiritual state that "transcends" the physical and the perceptive empirical only through a wise intuitive consciousness. With this we can conclude that the purpose of David Thoreau's citation was to show this intuition that transcends logic and physical state, leading each person to follow his own path.

David Thoreau probably was saying follow your own path, always listening to friends doesn't make the best of you.