What has won their hearts and minds is the simple gesture the American soldiers make as they patrol the streets of Iraq’s towns: the thumbs-up.

It’s common to see a dozen curious children gathered around an American tank or armored personnel carrier, trying to make friends with the foreigners and using the thumb pointed skyward.

“It means ‘OK.’ It means we are friends,” said Yousif Thamer, a 9-year-old third grader who lives in the poor neighborhood of al-Thawra in northeast Baghdad.

Nearly two months after the ouster of Saddam Hussein, the presence of American troops here is greeted with mixed feelings. Many Iraqis resent the U.S. occupation, but they know that without the Americans they would still be living under Saddam’s repressive rule.

It is the children who reach out most often to the Americans, running after their tanks and Humvees, whistling, clapping — and sometimes booing them.

In return, the soldiers smile back, say hi, take their pictures, and sometimes climb down from their vehicles to play soccer with them. Wearily, they endure the occasional jeers. The children are in awe of the Americans’ uniforms, tanks and guns, and the macho power they represent.

In al-Thawra, one of the most dangerous sections of Baghdad, the children are mostly barefoot and dressed in ragged, dirty clothes. They are famously untame, notorious for throwing stones at visitors and their cars.

But they love the soldiers — most of the time.

“They call us ‘baby,”’ said 7-year-old Zahra Karim, who also likes the Americans because they once gave her juice.

Do these children understand what the Americans are doing here? The answers are varied:

• “They are guarding places and companies so they don’t get looted,” replied Yousif.

• “They freed us from Saddam,” said 14-year-old Husam Sawadi.

• Karrar Karim, 11, said he doesn’t like them and doesn’t wave to them because “they are Jews.” He said his father had told him the soldiers “don’t pray. They are infidels.”

Yousif, though, wouldn’t stand for such talk.

“He’s lying,” Yousif said of Karrark. “He likes them; he gives them the thumbs-up. He does that every day.”

At an Army position in al-Thawra, a group of children climbed the fence to attract the attention of soldiers standing guard behind barbed wire.

“Look at the chickens,” one jeered at the Americans, using a word considered rude in Iraq. The others laughed and started jeering too.

“We’re used to this,” said one soldier, unbothered.

“They throw stones at us sometimes,” said Lt. William Earl, 25, from Tacoma, Wash. “They want us to play with them.”

He said the soldiers had made friends with most of the children, and knew the names of many.

But not everyone likes them. As a group of children were raving about the Americans, Karim Rahimeh, 37, limped into his al-Thawra home.

A former Iraqi soldier, he had three wounds in his right leg from when he was injured when the Americans bombed his barracks.

“Children don’t understand the crimes the Americans committed,” Rahimeh said. As he spoke, the children gathered in the yard, including two of his daughters, listening quietly.

Turning to the children, he said: “I am telling you, Americans are criminals.”

But Zahra, Karimeh’s seven-year-old daughter, was unmoved.

“I still like them,” she said. “They wave at us and do this,” she said, giving a thumbs-up.

But there are angrier voices too. In central Baghdad, a splash of graffiti, signed by something called the Baath Brigade — apparently a remnant of Saddam’s ruling Baath Party — gives an ominous warning:

“We shall cut off the hands that wave to the Americans.”