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":
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