Baldwin opens the third paragraph with the arresting statement that "any Negro who is born in this country and undergoes the American educational system runs the risk of becoming schizophrenic." This is clearly not intended literally. Schizophrenia is a chronic mental disorder which is rather rare. Some of its effects, however, are widely known and to mention it is to make a powerful emotional impression.

There follows the device of personifying the entire culture as a source of negative messages about the worth of African-American children. The child is told that "his ancestors were happy, shiftless, watermelon-eating darkies" and that their only value lay in their devotion to their white masters. Presumably no one has literally told the children this, but it is a crystalized version of what Baldwin believes is the dominant view.

The next paragraph is full of images showing how the black child is oppressed by the unfairness of the world without knowing quite what is going on. This culminates in the hurt and mystification of the child whose "father or mother slaps him and drags him to the back of the bus" when he sits down in the front. The appeal to pathos here is that the emotions of the child are the focus and vocabulary is emotive. Baldwin speaks of weight and menace, which the child feels without understanding the source.

The pathos in paragraph five comes from the excitement of the child on a day out at the beginning degenerating into the squalid everyday conditions which must be surmounted. To get into the atmosphere of normal childhood, the African-American child must pass through "the pimps, the whores, the junkies – in a word, the danger of life in the ghetto." This is one of the strongest appeals to pathos, since it presents such a strong contrast between the ideal atmosphere of a childhood treat and the grim reality for the child Baldwin describes.

Actually, the term "appeal to pathos" is a tautology. The Greek term "pathos" itself means "an appeal to the emotions," and is a type of persuasive rhetoric described by Aristotle. It may be accompanied by ethos (appeals to ethics) and logos (appeals to logic).

In paragraphs 3-5 of Baldwin's speech, there are several examples of pathos. Note the mention of "the stars and stripes" in paragraph 3, a sentimental way of referring to the American flag. Americans, more than people in most other nations, hold their flag in extremely high esteem and this phrase conjures thoughts of what it is meant to represent. To be "born in the shadow" of that flag, then, creates an image of a black child who is oppressed, rather than uplifted, by this great symbol. The pathos here draws upon the listener's attachment to the flag and its values; it is an emotional appeal to the listener in the hope that they will feel moved by the way the black children of the US are being failed. Likewise, the quotation "liberty and justice for all" prolongs this appeal. In the same paragraph, Baldwin seeks to provoke an emotional response--in this case, discomfort and, probably, shame--from the use of offensive terms to describe black people: "watermelon-eating" and "darkies".

In the following paragraph, the pathos is built upon the listener's presumed fondness for children, and attachment to the idea of childhood innocence. Children, Baldwin says, "look at everything, look at each other, and draw their own conclusions." The listener recognizes this depiction of the innocent, trusting child and, persuaded by the rhetoric, should then examine their own prejudices. Later in the same paragraph, the innocence of the child is contrasted to the black people whose shoulders bear "a terrible weight" which "menaces" the aforementioned innocent children. The imagery forces the listener to imagine the optimistic child confronted with this "menace," and the listener is moved by the picture.

Pathos is defined as an appeal to the audience's emotions and is considered a persuasive technique. The author of a speech uses images or stories that his /her audience can sympathize with in order to win their approval for his/her main argument.

"A Talk to Teachers," published in 1963, was originally delivered as a speech, "The Negro Child - His Self Image", the same year. In the paragraphs that you mention as well as throughout the speech, Baldwin appeals to his audience using both techniques associated with pathos: using metaphor or storytelling and the overall emotional tone which conjures up images with which the audience can relate to. Paragraph 3 starts with a strong emotional sentence

any Negro who is born in this country and undergoes the American educational system runs the risk of becoming schizophrenic.

Baldwin explains his powerful statement by making more references to images that he is certain will have a strong hold on his audience as they convey America's national myth of freedom and social mobility. In his second appeal to pathos, he says that, on the one hand, the African American child is born "in the shadow of stars and stripes," a symbol that guarantees liberty for all and the possibility to improve one's social standing ("He is part of a country in which anyone can become president, and so forth"). Yet, on the other hand, he is also consistently reviled by mainstream America as part of a race with no cultural or social heritage:

his past is nothing more than a record of humiliations gladly endured.  He is assumed by the republic that he, his father, his mother, and his ancestors were happy, shiftless, watermelon-eating darkies who loved Mr. Charlie and Miss Ann, that the value he has as a black man is proven by one thing only – his devotion to white people.  If you think I am exaggerating, examine the myths which proliferate in this country about Negroes.

 In this quote I have italicized what I consider another appeal to pathos: the institution that Americans often cite as a source of national pride, that republic which they built in opposition to the colonial monarchy, is the very source of stereotypes and discrimination against African Americans. Baldwin's implicit statement here may also refer to the trauma of the Civil War: what was the use of the Civil War if the state continue to perpetrate the stereotypes that were part of the slave-holding culture of the South?

In the last paragraph mentioned in your question, Baldwin makes his appeal to pathos by telling a story, reminiscent of his own childhood experiences, to prove how an African American child who ventures outside his neighborhood into the city understands his inferior and unequal condition.