Proctor refuses to sign a false confession in the end, refusing to set an example of weakness of character for his children. If he were to sign the confession, he would be allowed to live but would live as a moral failure, condemning others to die in his place. He cannot accept this.
Proctor refuses to do one thing that has two implications. He refuses to sign his name to a confession of witchcraft that will be publicly displayed. If he were to do so, he knows that the name of Proctor would be besmirched for future generations including his own sons. So, in a very literal sense, he is refusing to allow Danforth to tarnish his family's name. But Proctor is also concerned about teaching his sons to "walk like men in the world". This quote has more to do with the meaning behind Proctor's actions than the literal discussion of his name. If he confesses to witchcraft, he is taking the 'easy way out', so to speak. He would be tarnishing his own character by lying and this action would be the heritage that he left to his sons. He would be teaching them that taking the easy way out is more important than being a man of character, honor, and integrity. As the play tells us, Proctor does decide that character is more important and he proves this by choosing "goodness" and martyrdom.
It's fun to see the difference between the grade 9's answer and the grade 11 editor's answer