Why does Attean's grandmother change her mind about Matt in Chapter 19 of The Sign of the Beaver?

Prior to Chapter 19, when Matt attended a feast at the Indian village after helping Attean kill a bear, Attean's grandmother was not happy with his presence. Attean's grandmother hated the white man because her daughter was killed by them when Attean was young.

When Matt found Attean's dog wounded and caught in a trap, he attempted to save the dog and injured his hand. He turned to the Indian village for help, but Attean and the men were gone hunting. Attean's grandmother was told the story of how Matt helped save Attean's dog. She was impressed that a young white boy would risk harm to save an Indian's dog. This caused Attean's grandmother to change her mind about having him at the village. She sent Attean to the cabin to invite Matt back to the village.

Why does Attean's grandmother change her opinion about Matt in The Sign of the Beaver?

Attean's grandmother changes her opinion about Matt because of the trouble he went through to save Attean's dog. Attean says,

"My grandmother very surprise white boy go long way for Indian Dog...She say you welcome."

Matt had found Attean's dog caught in a white man's cruel trap. He tried to free the panic-stricken animal, but had been unable to, and had badly gashed his hand in the process. Not knowing what else to do, Matt ran to the Indian village for help. Finding that both Attean and Saknis had both gone hunting, he asked for Attean's grandmother in desperation. Attean's grandmother, who, for good reason, hated all white men, greeted him with coldness and suspicion, not understanding his language nor why he was there at the village. Fortunately, Attean's sister was present, and, understanding a little English, translated Matt's words to her grandmother, and begged her to allow her to help Matt save the dog. The grandmother, shocked that a white boy should be so concerned about an Indian dog, relented, but first demanded that Matt stay so that she could tend to his injured hand. Insisting that Matt's hand could get infected if not properly treated, she washed it clean with clean warm water, scooped "a pungent-smelling paste and spread it over the wound," then bound his hand with a clean bandage. The grandmother then gave Attean's sister permission to go with Matt, and the two were able to save the dog.

Two days later, Attean announced that his grandmother had invited Matt to come to the village for a visit. She had been impressed with his obvious compassion for her grandson's dog, and her attitude for the white man in was somewhat softened. Because Matt, in her eyes, took the first step in showing kindness, she was able to look at him at an individual, rather than just as a member of the people who had so hurt her own (Chapter 18 - 19).

Why does Attean's grandmother treat Matt's hand in The Sign of the Beaver?

While it is true that Attean's grandmother despises the "white man," she decides to treat Matt's hand after he hurts it trying to free a dog from a trap. There are several factors that worked together and allowed her to come to the decision to help Matt despite her negative feelings about his people. 

The dog that Matt attempted to help out of the trap belonged to Attean. This impressed his grandmother because Matt had no obligation to help the dog when he came upon it while walking in the woods. In addition, Matt and Attean's sister, who translated for Matt when he insisted upon seeing Attean's grandmother, pleaded with her at length about the situation. They needed Matt to be able to return to the trap with help to free the dog. Lastly, it was mentioned that if the wound were not fixed, it may get worse due to poison that may be in the wound.