The government has only lost a vote of MPs five times since 2010, and losing on the bedroom tax last month may prove particularly telling. David Cameron's defeat is partly political - a flagship Coalition policy in the balance, Lib Dem Ministers voting against him and Tory MPs failing to turn up, just a week after the first of their number defected to UKIP.

But the 304-237 Parliamentary vote is also important because this backbench bill now has chance of becoming law, and changing a government policy to an extent that very few would have expected when the bedroom tax was first brought in.

Most remarkably, the bedroom tax combines two subjects which pollsters tell us are perennially unpopular: social housing, and housing benefit. Less than one in six people now live in social housing, and only around a quarter would be happy to do so because of the stigma attached to the tenure. 

Two-thirds of people believe that we spend too much on housing benefit, on immigrants and families with too many children.

Despite this, as constituency MPs we were bombarded by campaigners ahead of the debate, urging us to vote for the bill. A national newspaper ran a dedicated campaign. 

A recent YouGov poll revealed that half of the public now say they oppose the bedroom tax, with only 41% supporting.  Against the odds, the campaign to end the bedroom tax has had striking success in mobilising people to join the fight to abolish it, and has gradually won the argument with the wider public too.

As social landlords, councils are at the forefront of managing the effects of the bedroom tax. Council tenants are the group most likely to be receiving housing benefit, and the bedroom tax is part of the reason why welfare reform is one of the biggest challenges facing local authorities over the next few years. So what can local government learn from the campaign against the bedroom tax, about ways to get the public and Parliament to listen?  Three things stand out.

First a clear name for the problem - the ‘bedroom tax'. You don't have to have a background in the complexities of housing benefit or social housing allocations to understand the nature of it and why it matters.  A campaign to reverse the CPI-uprating of LHA may be more important to tenants and landlords in the long run than the abolition of the bedroom tax, but if MPs and the public don't understand it and can't remember it, they won't support it.

Second a range of organisations coming together to campaign. Individual councils, the National Housing Federation, housing associations, tenants' groups, trade unions and others were able to provide not just a national campaign, but also area-specific action focusing on the number of people affected in their locality. Motions were passed at councils and petitions handed in. MPs were therefore presented with a range of groups and organisations with a united message and the first-hand evidence to back it up.

Third, clear argument backed up by real-life stories. The central case that the bedroom tax placed impossible demands on tenants by asking them to downsize to homes that just didn't exist was brought to life by the stories of people affected. Every MP in the country has desperate constituents driven to the very brink by the bedroom tax payments.

So when we came to debate this in Parliament, almost without fail every MP had a local story to tell – the young person pushed into the hands of loan sharks, the carer and his disabled wife told they had to do without essential extra space for medical equipment, the family with school-age kids facing eviction. It is these tales that have helped win the argument with the public and, on this topic at least, overturn the unsympathetic way in which social tenants are normally portrayed.

So far, so good then for the campaign against the bedroom tax. The backbench bill still has to pass stages in both the Commons and Lords to become law, but the highly convincing, broad-based fight against the bedroom tax means the bill to make significant changes in with a chance. Lest campaigners congratulate themselves prematurely though, remember one fact: Even if the bedroom tax were totally scrapped today, this would reverse less than 7% of the housing benefit cuts in this Parliament.  For councils and for all of us, there is more campaigning to do.

John Healey is former local government minister