President Barack Obama just stopped the bleeding. Two weeks ago, the first debate in Denver seriously hurt the president, who saw his six-point lead cut to a statistical dead heat in national polls. Going into the second debate, Gov. Mitt Romney owned the momentum, with his once ailing campaign slicing through the president's edge.

Tuesday night at Hofstra University, however, was a contentious battle for undecided voters in swing states.

Obama attempted to sway undecided voters by returning to the narrative that had him leading the race before his listless debate on Oct. 3. He painted Romney as an ally of the rich and powerful who doesn't have middle-class interests at heart. He underlined Romney's tax cuts for the wealthy -- even highlighting the former Massachusetts governor's turn on "60 Minutes," in which he said it was fair for him to pay a lower tax rate on his $20 million income than a nurse making $50,000.

This was all a lead-up to the final punch, when the president cited the now infamous video of Romney writing off 47 percent of the country as government-dependent "victims" who believe they are "entitled" to health care, food and housing.

Polling done by Public Policy Polling in Florida, Colorado and Wisconsin -- key swing states -- after that 47 percent remark first drew wide attention revealed two important things: 1. At least 89 percent of likely voters were familiar with the video; and 2. The vast majority were left with a negative impression, making them less likely to support Romney.

So it was smart strategy for Obama to highlight it in his closing remarks.

Romney was also dogged in his pursuit of undecided voters at Hofstra, raising questions about the president's ability to create jobs and correct the economy. He hammered home his five-point economic plan, targeting China as a currency manipulator and vowing to free up federal land for domestic energy. Undecided voters list the economy as their top concern, and polling shows that voters view Romney as more capable of lowering the federal deficit, which he repeatedly cited as outrageously high.

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But statistically, we need to remember that it's misleading to treat undecided voters as a monolith. The common wisdom is that undecided voters are white, working or middle-class, with some or no college education, who don't pay close attention to policy issues. However, undecided voters account for only 6 to 12 percent of voters. That means that any polling is subject to a large margin of error due to the small sample size, according to Mickey Blum, director of Baruch College Survey Research.

Which is all to say, we don't really know what undecided voters think. We know that an undecided voter in Florida or Nevada -- home to large Hispanic populations -- likely appreciated Obama's strong defense of the Dream Act. And we know that Romney's opposition to the assault weapons ban and any further regulations on firearms probably spoke to the undecided voter in Colorado. And both candidates may have scored points with Colorado, Pennsylvania, Ohio and Virginia undecided voters, with their strong support for coal.

What we do know for sure is that in a tight election, turnout matters. So it makes sense that both candidates were passionate about firing up their bases. Whether undecided voters react with the same enthusiasm as the partisans, though, is anyone's guess.

Alexis Grenell is a Democratic political communications strategist.