Malala writes, “In Pakistan when women say they want independence, people think this means we don’t want to obey our fathers, brothers or husbands. But it does not mean that. It means we want to make decisions for ourselves” (219). What is the difference? In the US, we put great stock in the idea of independence—how is our understanding of independence similar or different?

The Gender Inequality Index ranks Pakistan a lowly 150th out of 160 participating countries. The United States is ranked thirteenth, so the status of women varies enormously between the two nations. Although America ranks much higher than Pakistan, it is not even in the top ten, so it is obviously an imperfect paradigm for gender equality.
Born in Pakistan in 1997, Malala Yousafzai was nearly murdered at the age of fifteen by an extremist who opposed girls' education. Malala had made a name for herself by speaking for Pakistani girls' right to an education. After her recovery from near-fatal wounds, she continued her campaign from abroad. In 2014, she won the Nobel Prize.
Malala states, “In Pakistan when women say they want independence, people think this means we don't want to obey our fathers, brothers or husbands . . . [but] It means we want to make decisions for ourselves." She is challenging the idea that the movement for women's liberation is rooted in disrespect for all men. Malala is a staunch proponent of women's rights and personal freedoms. Another quote from Malala along these lines is the following: "I believe it's a woman's right to decide what she wants to wear and if a woman can go to the beach and wear nothing, then why can't she also wear everything?"
Pakistan is an extremely conservative Islamic nation. In her memoir, Malala does not want to sound like an extremist. Also, she has great respect for her father, which might be why she emphasizes that women don't necessarily want to disobey men. In other words, it is important to put her words in the context of her ultra-conservative nation. In view of her tragic personal history, her caution is understandable.
http://hdr.undp.org/en/content/gender-inequality-index-gii

The point Malala's getting at is that just because a woman or girl wants to make her own decisions in life, it doesn't necessarily mean that she's being willfully disobedient. Inevitably, there are some decisions that should only be made by parents and other adults in authority. Likewise, there are decisions that should only be taken by individuals themselves, no matter how young they may be. In the West, one such decision would be whom to marry. In more traditional countries such as Malala's native Pakistan, however, marriages tend to be arranged by the families of the bride and groom.
The American understanding of independence tends to be conceived in negative terms. That is to say it's independence from something or someone. The United States was founded on a desire for independence from the British. This would allow the American colonists to make their own decisions about how they should run their affairs. What Malala is articulating in the above extract is a positive form of independence; the independence to do something in life. Unfortunately for her, Pakistani culture tends to confuse this more positive understanding of independence with the more negative American variant, thus perceiving it as a threat to society and the traditional family structures on which it's based.