Back in 2001, when Congress was first considering the No Child Left Behind Act, former-U.S. Secretary of Education Margaret Spellings and U.S. Sen. Lamar Alexander, R-Tenn., were on more-or-less the same page when it came to a strong, federally driven accountability system.

But since then, Alexander (who served as President George H.W. Bush’s secretary of education) and the Republican party have been moving farther and farther away from the idea of a federally driven accountability system. Spellings, meanwhile, has stayed true to the law she once compared to “Ivory Soap—it’s 99.9 percent pure.” UPDATE: To be clear, Alexander wasn’t in the Senate at the time NCLB passed, but in the early 2000s he was very positive about the NCLB law, and supported a bill that would have left its accountability system largely intact.

Can these two find common ground? That’s what the Thomas B. Fordham Institute, a think tank in Washington with long-standing relationships with both Alexander and Spellings, tried to figure out today. If you’re an edu-politics nerd, you should absolutely check out the video here.

The upshot? No big consensus on where the GOP should go when it comes to ed policy—but there was some very interesting back-and-forth.

Both Spellings and Alexander said they generally like the Obama administration’s overall strategy of offering states competitive grants in exchange for embracing certain reform priorities. And they are both pretty down on the Title II program, which gives grants to states and districts to improve teacher quality. They both think it’s a huge waste of federal resources.

But Alexander strongly supports the nearly $300 million Teacher Incentive Fund, which offers grants to districts to create pay-for-performance programs. It got started under Spellings, and Secretary of Education Arne Duncan has sought to expand it.

And while Spellings may like the general idea of Race to the Top, she wants to see actual results. Expanding charter caps, which many states did to get a grant, is one thing, she said. But where are the new charter schools?, she asked.

Neither of them are huge fans of the Education Department’s NCLB waivers, but for totally different reasons. Spellings thinks states have set the bar too low, particularly when it comes to holding schools accountable for the performance of students who are members of particular subgroups. And Alexander thinks the waivers pretty much trampled on Congress’ authority.

Alexander also gave a big thumbs-up to presumptive GOP nominee Mitt Romney’s proposal to let Title I grants for districts follow the child. Spellings, who jumped off Romney’s advising team because she wasn’t so happy with the policy direction he was taking, kept silent on the issue.

Both were asked whether they thought members of the tea party, many of whom basically see no role for the federal government in anything K-12-related, could be persuaded to go along with any bill that keeps the Education Department open for business.

Alexander joked that “he was tea party before tea party was cool.”

But Spellings sounded a note of caution. She’s worried that the tea party’s anti-Education Department stance may be hurting the GOP’s electoral chances, particularly with women and Hispanic voters. Alexander agreed that Republicans did much better on K-12 when Bush talked about it a lot.

So why isn’t an Elementary and Secondary Education Act renewal done yet? Alexander, indirectly, appeared to place some of the blame on the White House for not making it a top priority.

“President Bush and Margaret devoted full attention,” to the issue until it was done, he said.

The whole event was, of course, total Twitter catnip. For you non-Twitter-savvy folks, Fordham has a great tweet-wrap on its website. Or you can search the hash-tag #NCLB10 to read some of the back-and-forth.