Barack Obama and George W Bush are in most respects political polar opposites. But as incumbent presidents, they have one glaring trait in common. They can't admit they've made mistakes.

Bush, to be fair, makes some amends in his new memoir, Decision Points, conceding that his handling of Hurricane Katrina left something to be desired, and that he deployed far too few troops in Iraq after the overthrow of Saddam Hussein.

But contrition on the part of Bush the author comes too late to expunge the memory of Bush the president being asked by a reporter, as his poll ratings tumbled, if he could think of any mistakes he had made in office. A long and awkward silence followed, before he mumbled something about a few second-tier judicial appointments.

Cut to last week, and the morning after the Democrats' biggest mid-term drubbing since the second world war, and Obama's press conference in the White House. Of contrition and repentance there were next to none. Insofar as he was responsible for his party's "shellacking", it was because he had not explained his policies properly. But that was it. He might be advised to issue a somewhat louder mea culpa if he wishes to avoid being a one-term president.

Such a fate, however, is anything but pre-ordained. In 1994 Bill Clinton took an almighty mid-term beating, only to be comfortably re-elected two years later. There is no a priori reason why Obama should not do the same. This vote was a protest vote, against the president and against "big government", but also against the wretched economy and the failure of Congress to correct it.

There were understandable human reasons for Obama's reluctance to admit error. From the moment he came to office, he has had to face unrelenting, often spiteful Republican opposition to his every initiative. He inherited the toughest in-tray in living memory. And, quite simply, it would take anyone a while to digest a thrashing like 2 November.

But a certain pattern has emerged in American electoral politics of late, not unlike those mathematical sequence questions in IQ tests. In 2006 the Democrats won a smashing mid-term victory, in 2008 they recaptured the presidency. In 2010 the Republicans won a smashing mid-term victory, in 2012...?

These next two years will be a voyage of discovery for us all.

Undoubtedly, the expectations Obama raised in 2008 were too high. As the Bush presidency limped to its inglorious end, the young Democrat newcomer seemed to embody a rebirth of the republic. His obvious intellect, his thoughtfulness and composure made us forget how inexperienced he was.

He had, and still has, an uncanny confidence in his own ability. But all we had to go on when he ran for president, however, were eight years in the Illinois state senate, and two charisma-drenched but otherwise unremarkable years as a US senator before he hit the campaign trail.

People could, and did, mock the younger Bush as a political hick – but he had twice been elected governor of what is now the second-most populous state in the union. As for Clinton, he had been governor of Arkansas for a dozen years. Everybody knew he too had some game, but he suffered a crushing defeat in his first bid for re-election, and was forced to rethink his approach to regain his job.

The experience offered valuable pointers to how Clinton handled his 1994 humiliation, which forced him at one point to plead his continuing "relevance" to the governance of the country. He changed his game, and went on to secure a second term. The greatest question in American politics right now is, can Barack Obama do likewise and change his?

Last Wednesday morning, he sounded chastened, but basically unapologetic. His policies were correct; the problem, Obama seemed to imply, was the failure of voters to understand that. His punishment was unjust, even incomprehensible, except in terms of a failure to communicate. The harsh reality, however, is that the playing field has been transformed – to the extent there are even murmurings he may be challenged from within his own party for the 2012 Democratic nomination. That would be a calamity. Every recent sitting president who has faced a significant primary opponent has lost the general election.

To avoid that fate Obama has to regain a skill he has lost, and demonstrate others he has yet to show. The vanished art is eloquence.

As for the unproven skills, they include a genuine readiness to compromise and a deftness in exploiting cracks between the Republican Congressional old guard and fractious Tea Party newcomers. Nor would Clinton's Ali-esque tactic of political 'rope-a-dope' go amiss – taking everything his opponents threw at him and luring them into a fatal overconfidence. Best of all, he could start admitting the odd mistake.