Let us call the two sides inputters and outputters. The inputters favour more money and smaller classes. The outputters favour, well, less money and larger classes, I suppose. However, while the inputters are disarmingly precise about the class size they would settle for (17), the outputters never say whether they want classes of 30, 40 or 100. All they know is that we should stop pouring money into education and tell teachers to pull their socks up.
The inputters cite Tennessee's Project Star, in which 11,600 pupils were randomly assigned for their first four years of schooling to normal-sized classes (which means about 22 in the United States, you will be jealous to hear) and to smaller classes (about 15). Test scores in the smaller classes came out significantly higher. Project Star dates from the mid-1980s, and was quoted by Labour in support of its 1997 election pledge - still not 100 per cent fulfilled - to abolish all classes over 30 for infants.
The outputters cite population studies. Broadly, countries that spend most on education do not get the best scores in international tests; schools with low pupil-teacher ratios often do worse in league tables; and people who have been educated in small classes do not - when the effects of school type are stripped out - do any better in life.
One inputter quotes Galileo: the outputters have "a hundred dray horses" to support their argument. But population studies rarely prove anything decisively. Is there a clear association between a country's alcohol sales and its convictions for drunkenness? Probably not; but if I failed to find one, I would not then conclude that alcohol has nothing to do with getting drunk.
The inputters have recently invaded the outputters' territory, with an analysis based on all children born in England and Wales in one week of 1958. This concludes that class size has a substantial effect on the decision to stay on at school and, through that, on wages in later life.
Every time class size is reduced by one, the boys get an extra pound;683 each in lifetime earnings, the girls pound;493.
But the "Barbary steed" (to quote Galileo again) that carries the inputters' case is still the Tennessee Star project. This is a rare example of a social science experiment that follows the principles of medical research: it treats one group, while leaving another group untreated and using it as a control. But it is the only steed in the inputters' stable - it can run fast, but carries a heavy burden.
Yet the inputters have two arguments that stare them in the face. Both involve turning the outputters' own convictions against them. First, if it is true that what really makes a difference to school performance is teacher quality, then the case for smaller classes is already made. Nobody denies that teaching smaller classes involves less stress and more job satisfaction. Teaching would be a more attractive career - and have a better chance of attracting smart people - if recruits were guaranteed classes of 17.
The second, even better, argument is that parents actually pay for small classes. Why would they do that if there is no benefit? The outputters are mostly market economists who believe in the infallible rationality of consumer decisions. So if reducing class sizes is a waste of money, the fee-charging schools should long ago have met customer resistance and been forced to employ fewer teachers and lower their fees.
The inputters should abandon their regression analyses. With these simple arguments, they and their single Barbary steed can rout the outputters and their dray horses once and for all.
Peter Wilby is editor of the New Statesman