Updated 12/10/12 – 5:45 p.m.

CHICAGO (CBS) — Mayor Rahm Emanuel responded gently on Monday to criticisms of his crime-fighting strategy leveled at him last week by Cook County Board President Toni Preckwinkle.

“Toni and I have an immense agreement that relates to what we’re talking about today. … My number one concern … or top priorities, however you want to say it, is public education and public safety,” Emanuel said Monday.

WBBM Newsradio Political Editor Craig Dellimore reports, though she quickly backed down on her comments, Preckwinkle raised eyebrows on Thursday when she slammed Emanuel and Police Supt. Garry McCarthy’s law enforcement strategies.

“Clearly this mayor and this police chief have decided that the way in which they are going to deal with the terrible violence that faces our community is just to arrest everybody,” Preckwinkle said last week at the Union League Club.

LISTEN: WBBM Newsradio Political Editor Craig Dellimore Reports

“We have five percent of the world’s population … we have 25 percent of the people in the world who are in jail or prison,” she added. “A lot of it relates to the way we treat drug offenders. We’ve decided in this country that we’re going to treat drug offenses as a criminal matter, instead of a public health issue. But a lot of it is just the cumulative effect of this hopelessly destructive and stupid ‘get tough on crime’ binge that we’ve been on for the last 30 years.”

Speaking to reporters shortly after that criticism, Preckwinkle toned down her comments, and on Monday the mayor said he and Preckwinkle are on the same page when it comes to fighting crime.

“We will do what we can to give our kids a good education, so more kids see college in their future. We’re going to make sure our kids have an after-school activity — so that when crimes are committed between the hours of 3 and 6 [p.m.], and more kids are victims – there’ll be more kids in after-school activities,” Emanuel said. “But we will also make sure that there’s the full force of the law that’s there and available.”

The mayor said he’s committed to doing more than simply arresting criminals

“When it comes to public safety, I’ve always believed in … putting more police on the street [and] getting kids, guns, and drugs off the street,” Emanuel said.

There used to be an open-door policy, literally, between Emanuel and Preckwinkle. The two met all the time. Now, it might be more than symbolic that the door between their Fifth floor offices at City Hall and the County Building is closed.

They looked like the odd couple, but their policies and politics were very much alike. Both won big when they ran for office, and then promised to work together.

At first, they did, but after Emanuel’s approval rating declined and Preckwinkle’s increased, their relationship might have changed from allies to rivals.

“I’m going to leave it to politics, and I don’t really care about that,” he said, trying to downplay the conflict between the two.

So did Preckwinkle, to a point.

Shortly after criticizing the mayor’s crime-fighting strategies last week, she backtracked a bit.

“The mayor and I jointly talked about these initiatives,” she said Monday. But she refused to take back last week’s crime fighting criticism.

“Policing alone will not address the social issues that face our communities,” she said.

Preckwinkle also criticized the Chicago Public Schools last week, in particular the district’s low graduation rates over the years.

“We have contented ourselves with a miserable education system that has failed many of our children,” Preckwinkle said last week.

However, she later said her comments about public schools were not directed at the mayor, but at society as a whole.

On Monday, Emanuel said he and Preckwinkle agree on education reforms that are underway.

“There’s nothing more important to dealing with what we have to deal with – both from an economic development, from job creation, to public safety – than an education,” the mayor said.

He said a longer school day is a major step in the reform process.

There are some who wonder whether Preckwinkle might be positioning herself to run against Emanuel two years from now, but a Preckwinkle spokesman late Monday afternoon left no wiggle room, saying: “absolutely not.”

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