According to the conventional wisdom about the 2010 midterm election, Obama-supporting Democrats in competitive House races were punished at the polls.

Political scientist Brendan Nyhan found that votes for health-care reform, cap-and-trade, and the stimulus package took a decisive toll on Democratic incumbents. Former Senator Evan Bayh says his fellow Democrats “overreached by focusing on health care rather than job creation.” They therefore need to tack right—with a conservative agenda of tax breaks, deficit reduction, and entitlement reform—in the run-up to the 2012 campaign.

But Democrats, especially those who are unhappy with that agenda, should look at the November results more carefully before following Bayh’s advice.

Consider the victorious campaign of incumbent Representative Martin Heinrich. The New Mexico Democrat defeated a well-connected Latino challenger in a disproportionately Hispanic state that simultaneously elected a Latina Republican governor by an eight-point margin. What is more, the New Mexico freshman—who voted in favor of health-care reform, cap-and-trade, and the stimulus package—beat his Republican challenger in a district that until 2008 had never elected a Democrat.

How did he do it? Heinrich apparently took seriously former House Speaker Tip O’Neill’s dictum that “all politics is local.” He returned to his Albuquerque home during more than 80 of his first 85 weeks in office and participated in more than 350 community and constituent meetings. Observers report that they’ve never seen a first-term representative spend as much time in his or her district. When he assumed office, Heinrich reaffirmed his dedication to the provision of high-quality constituent services, and he quickly made good on his promise. “People should have access to their government,” he explained to the New Mexico Independent, “and constituent services are the number one way to make that happen.”

Constituent services help people access their government—federal organs such as the IRS, the Department of Veterans Affairs, and Social Security Administration—and also give officials access to the people, including information about their needs. Heinrich says his focus on services explains his first major legislative decision: to support credit-card reform. Soon after Heinrich assumed his post, his office was flooded with complaints about credit-card companies’ new interest rates, fees, and due dates. He responded by voting in favor of the Credit Card Accountability, Responsibility, and Disclosure Act of 2009 as well as subsequent amendments that were designed to head off preemptive rate hikes. “It was easier to get the local media to cover that than many of the aspects of the health care debate,” Heinrich told Real Clear Politics, “it really resonated with a lot of my constituents.”

Credit-card reform was relatively uncontroversial. It passed both houses of Congress with strong bipartisan support and was signed into law by President Obama in the spring of 2009. However, by keeping his finger on the pulse of his district, Heinrich was also able to “localize” the more controversial legislation that emerged during his first term in office, including health-care reform and the stimulus. For instance, he made sure that the health care–reform legislation included language reauthorizing the Indian Health Care Improvement Act, and thus brought tangible benefits to tens of thousands of his constituents. And he secured stimulus funding for workforce and product development in New Mexico’s growing clean-energy sector, thereby addressing demands for job creation and environmental protection simultaneously.

Such victories helped Heinrich in the November election. He raised money from a broad cross section of his district—from high-tech entrepreneurs and professionals to working families and unions—and mobilized hundreds of volunteers to make phone calls, go door-to-door, and do whatever else was necessary to ensure victory. In the closing weeks of the campaign, facing a sudden onslaught of attack ads financed by independent groups, Heinrich asked his supporters to dig a little deeper. They responded with a massive get-out-the-vote effort that put him over the top on Election Day.

Heinrich’s reelection is striking but not unique. A few other first- and second-term Democratic representatives, including Bruce Braley (IA), Tim Walz (MN), and Chellie Pingree (ME), also managed to reconcile their support for Obama’s agenda with victories in competitive reelection campaigns. Pingree even increased her margin of victory over 2008.

The exceptions are almost always harder to explain than the rule, but we should look carefully at the strategies the winners pursued. Do constituent services, grassroots mobilization, and the localization of national issues provide progressive Democrats a recipe for victory?

The answer won’t be revealed in a brief look at a single campaign in a small state. But the payoff of a more granular analysis could be enormous. The approach advocated by former Senator Bayh would be disastrous for the future of the country, and there is no solid evidence that it would result in electoral gains. Maybe Representative Heinrich is onto something.