Walker, at campaign stops in Milwaukee and Middleton, hammered Burke for not supporting expansion of the private school voucher program, which makes taxpayer money available for students to attend private school.
Walker also said Burke was hurting the Madison district by not arguing for implementing Act 10, the law that effectively ended collective bargaining for most public workers and also forced them to pay more for pension and health care benefits.
Burke, in a conference call with reporters, responded by saying the best way to tackle the achievement gap is fairly compensating teachers. She defended Madison schools for negotiating new employee contracts while the union law was being challenged in court, and she said she would work to repeal the statewide voucher program outside of Milwaukee and Racine, saying there was "no basis" for expanding it.
"The key to improving student learning, that anyone who really looks at education knows, is the quality of the teacher in the classroom," she said.
Burke supports the higher pension and health care contributions required under Act 10 but has said she wants to restore collective bargaining for public workers.
Burke joined the school board in the spring of 2012. In June, the board unanimously approved a new collective bargaining agreement for Madison's teachers. That makes it the only teachers' union in 424 public school districts in Wisconsin with a collective bargaining agreement for both this year and the next, the Madison union said after making the deal.
The Madison district could have saved millions if Act 10 were enacted, money that could be put back into classrooms, Walker said.
In Madison, about 53 percent of black students graduated, compared with 87 percent of white students, based on 2013 data on the state Department of Public Instruction website. In Milwaukee, the only larger district, 59 percent of black students graduated. Statewide it was 65 percent.
Walker called the difference between Madison and Milwaukee's black student graduation rates "shocking to many people."
Walker also said Burke could also help address the achievement gap by supporting growth in the private school voucher program. Walker and other supporters of the program say it gives students in struggling schools another option, while opponents say voucher schools are unaccountable and divert money from public schools.
The 2-year-old statewide program, which operates separately from the older programs in Milwaukee and Racine, currently has an enrollment cap of 1,000 students. It is offered in 26 private schools statewide.
Walker said that he wants to grow the program, but he doesn't have a number in mind for how large he would raise the enrollment cap.
"We're going to look at it in stages," Walker said. It makes sense to do it that way, like Milwaukee and Racine did, "so capacity could be met," he said.
Voucher backers have been strong financial supporters of Walker's, while Burke is benefiting from unions and opponents of the program.
In the 2012 recall, Walker got $2.2 million from voucher supporters. In the first six months of this year, Walker's campaign got $74,500 from voucher backers, according to an analysis by the liberal group One Wisconsin Now.
The Wisconsin Education Association Council, the statewide teachers union, has funneled $1.3 million to a liberal group that is running television ads attacking Walker.