All this is happening when school education everywhere is under enormous stress to meet the demands of growing needs and population increases.

Many interests, including political parties, pressure groups, churches and unions, are involved and internal party rebellions are matched by external pressures. It is testing the unity of political parties at federal and state level. Traditional relationships are also fracturing just as new alignments are being forged. The situation in the Federal Parliament remains so uncertain that both houses and almost all MPs remain relevant as the critical vote approaches.

The federal government is struggling to keep its own MPs, especially some Catholic ones, in order, while at the same time it is trying to stitch up a Senate majority for its proposals, either with the Greens or the crossbench. Labor has turned its back on the government's plans despite criticism from friendly circles that it should support something based on its own original ideas. The Greens' leaders are negotiating compromises with the government while also trying to quell dissension within their own ranks. The crossbench is negotiating, too, but for once it is largely out of the limelight.

The Australian Education Union and the National Catholic Education Commission, now led by former federal Labor MP Christian Zahra, are campaigning strongly against the government. This combination of forces, which is somewhat unusual, will reinforce Labor's resolution. The union's opposition must trouble the Greens, as they flirt with compromise, and the church's opposition must unsettle Catholic MPs in Coalition ranks as they consider jumping ship. Some of them, like veteran Kevin Andrews, are already unsettled anyway by the long-running internal leadership wrangling.

There are now so many Catholics in the Coalition party rooms – a long-term demographic shift along religious lines between the major parties since the 1970s – that such Catholic dissatisfaction threatens the Coalition government. Catholic education bodies with links to potential dissidents like Liberal senator Chris Back say they have not only not been heard but, unprecedently, have not even been afforded serious engagement by the minister.

The wholesale shake-up includes state-level politics and crosses political party lines. State governments remain the major funders of public education. In NSW, Liberal Education Minister Rob Stokes remains resolutely opposed to the new deal because NSW did very well out of the generous previous arrangement with the Gillard government. Yet the former, highly regarded, NSW education minister, the Nationals' Adrian Piccoli, urged acceptance of the new model because, while imperfect and still underfunded, it is heading in the right direction on fairness.

Piccoli's position – given a further twist because he, too, is Catholic – is a good place to begin in trying to understand the party politics, because he is frank about the weaknesses of the Turnbull-Birmingham scheme but nevertheless argues it will unwind some of the inter-sector unfairness presided over by the Liberals' John Howard and Labor's Julia Gillard.

He reckons Labor should accept the new scheme but pledge to increase funding to more appropriate levels when next it is in office, while both Labor and the Greens should reject the opposition of public sector teachers' unions. That is always a big step for a labour party and not one the Greens, keen to build up such links, would want to take either.

But Labor, including many of its own MPs with deep family ties to Catholic education, is following a different logic. First, it believes it would endanger its traditional advantage over the Coalition among voters on educational provision and therefore may weaken its chance of winning the next election. As a consequence, it would not have the opportunity to increase funding anyway because it wouldn't be in office. Second, as an opposition, Labor is not inclined to offer Malcolm Turnbull a win and must be emboldened by the government's troubles with the Catholic sector.

The Greens are driven by their leader Richard di Natale's expressed desire to be at the table. He wants to build a new reputation for his party as a parliamentary deal-maker, along Australian Democrats' lines. But he and his education spokesperson, Sarah Hanson-Young, are treading on dangerous ground, too. Remembering the damage that the Democrats' GST deal with the Howard government did to that party, they must have a convincing story to tell their members and supporters about concessions won. An unlikely alliance with the government against the wishes of education unions will be a hard sell and could split the party. The amended package would need to be demonstrably "Greener" than the original.

The government, if it holds together, may be driven back towards some further negotiations with crossbenchers, which usually involve the sort of special deals supposedly anathema to Education Minister Simon Birmingham. Anything is possible in a last-minute scramble. It's better to delay.

Whatever the outcome this week, there is unlikely to be definite closure on education funding. Negotiations will continue and many strong voices will demand to be heard.

The government remains behind in the polls and has stirred up a hornet's nest of disputation, which is sure to be an important factor in the next federal election.

John Warhurst is an emeritus professor of political science at the Australian National University. john.warhurst@anu.edu.au