Walt Whitman

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What does Whitman's quote in "Song of Myself" mean, and how does it get interpreted in Julia Alvarez's "Daughter of Invention"?

"He most honors my style who learns under it to destroy the teacher."

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This quote is from Whitman's poem "Song of Myself", and it means that there is great honor for the teacher when his students manage to surpass him. 

This is made more clear in the stanza as a whole, which reads: 

I am the teacher of athletes,
He that by me spreads a wider breast than my own proves
     the width of my own,
He most honors my style who learns under it to destroy the
     teacher.

"He that by me spreads a wider breast than my own proves the width of my own" suggests that a student who can use the teacher's methods to achieve even greater things than the teacher ends up proving the teacher's greatness in the process.

This idea is not new: in the 16th century, Isaac Newton said "If I have seen further, it is by standing on the shoulders of giants." This suggests that Newton's work would not have been possible without the great men who came before him. 

Whitman's quote expresses the same sentiment, but in the opposite direction: from the perspective of the teacher rather than the student. Whitman essentially claims that, if his students go on to surpass him, it is because they have stood upon his shoulders. Their success thereby proclaims him a 'giant'; a man of great significance to the world's outstanding achievements. 

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Walt Whitman was what we can an iconoclast, literally someone who breaks the icons. An iconoclast is someone who "seeks to overthrow traditional or popular ideas or institutions." In other words, he broke all the established rules of poetry writing--and some of society as well.

He introduced a new form of poetry: free verse. There is no set meter or rhythm or rhyme scheme. It is almost like an essay written out line-by-line. Critics, mostly those "teachers" he was talking about in the quotation, hated his work. They called it "artistic laziness and exhibitionism, looseness and vulgarity" and attributed his "pioneering to ignorance and ineptitude."

So what Whitman was telling us is that if we want to follow his style, we need to forget everything we've learned about how to write poetry.

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