The school system systematically fails to recognise the head teachers who make the biggest impact in improving pupils' chances, research suggests.
Heads who focus on short-term success, even at the expense of the long-term, tend to be rewarded by the system.
The study looked at 160 schools in England and is published in Harvard Business Review.
Leaders who build big, successful schools are less likely to be recognised.
The new research, produced by the Centre for High Performance, shared exclusively with BBC Newsnight and Schools Week, external, is based on internal administrative data from schools which covers everything from pay to timetables.
Four researchers had access to 160 secondary school academies in England - building a dataset that covered the tenure of 411 head teachers.
The research, by Alex Hill, Liz Mellon, Ben Laker and Jules Goddard, is being published in the Harvard Business Review, external. Their work found that heads tended to fall into one of five "types":
"Philosophers" are the largest group. They are heads who do not see themselves as managers, but try to lead by example as senior teachers. They are inspirers, who like to talk about pedagogy, in particular. They do not change much about the student body or the staffroom.
"Surgeons" are head teachers who act decisively to try to turn around schools. On arriving in a school, they exclude an average of around a quarter of the final-year students and drive resources into final-year students. They fire around a tenth of staff. They have a dramatic immediate impact.
"Architects" are careful planners. They work on improving standards of behaviour in schools as a first step before working on improving teaching. They value parental engagement, seeing themselves as working for their community. They only expel children for behaviour management, and slowly replace poorly performing staff.
"Soldiers" and "Accountants" are best suited to schools which need a financial turnaround. "Accountants" try to increase the size of the school as a strategy for improving the financial balance. "Soldiers" try to cut costs to meet the school's budget constraints.
The graph below shows the average annual change they made when the head was still in post. The vertical axis shows performance change in exam results, measured as a share of the children in the school getting five good GCSEs. Dots that are higher up mean the school has a faster average improvement rate. The horizontal axis is financial balance. Being further to the right means that the school tended to improve its financial position.
The clear finding here is that surgeons appear to make the most dramatic improvement in the short term, averaging around a 10% improvement per annum in exam results with only a slight loss of financial strength. "Philosophers" make marginal improvement on results, but not finance. "Architects", on these measures, make steady progress on both finance and results.
All of this research is novel - elements of it may get refined and improved. We need to look more closely at some of the detail - like why different sorts of schools might recruit different sorts of heads. But it is not the case that, for example, architects just take easier schools. And there is certainly enough here to suggest that some strategies being pursued by heads may be counter-productive, yet those strategies are ones which get the rewards. The work also raises questions about the structure of English education.
First, it implies that the incentive structure for English schools is much too short-termist. We honour heads who get short-term results and pay them more. Architects, who maximise impact for the maximum number of children over the maximum amount of time, are not recognised. If anything, they are penalised. The short time-horizon is the consequence of regulation about school turnaround, the attitude of Ofsted and the way academy chains have to demonstrate rapid progress in order to be allowed to expand. All of these things are easily fixable.
Second, we do not look enough at school size. We have, until now, measured schools by the proportion of children meeting certain fixed benchmarks. This has always created an incentive to exclude children. One unforeseen consequence of these new progress measures now being introduced is that those incentives will disappear. They may, then, reduce the number of children being removed from schools. But explicitly paying more attention to school size might also help identify heads who are quietly excellent.
Third, it implies that the training of heads is insufficient. It is extremely striking that the "philosophers", for example, are mostly English teachers, the surgeons are mostly PE teachers and the architects are mostly history and economics teachers. Schools Week, external has a lot more detail on that fascinating avenue. That observation brings out how far teachers work in silos surrounded by people who had similar academic and professional training - and how little we do to broaden their experiences as part of the process of turning them into heads.
Finally, it is surely excellent news that a lot of schools can be made to work better with the student bodies and staffrooms within them. While the problems it raises are fascinating, the research is optimistic. And it is exciting - for once, we are able to talk about how good schools work. Not just about types of schools.
Chris Cook, external is policy editor for BBC Newsnight. More on this story on Newsnight at 22:30 on BBC Two Thursday - or catch up afterwards on iPlayer