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Put a child of a cleaner from Shanghai or Singapore up against a scion of the western elite in a standardised test and guess who will come out top? According to the latest research, the western kids will trail their Asian counterparts by the equivalent of a whole school year.

This prompted another bout of anxiety of a kind that has become increasingly common since 2001, when the global Pisa survey of educational attainment was first published. Parents once drew comfort from steady improvements in school-leaving grades in places such as the UK. Confronted with evidence of how their children’s accomplishments compared to those of students in faraway places, many westerners have taken fright.

Next week Elizabeth Truss, a British education minister, will lead a fact-finding mission to Shanghai to try to find out what the schools there are doing right. Yet in their rush to copy the winning formula of high-performing countries in east Asia, politicians risk drawing the wrong conclusions. Schools in Shanghai are very different from those in Ms Truss’s constituency in southwest Norfolk. But not all of those differences play a role in Shanghai’s superior performance. Some are irrelevant. Some may even be harmful. And some will be idiosyncratic features of the school she happens to visit, rather than representative of the system. It is easy to point out how a good school differs from a bad one, and conclude that you have found the secret to high achievement – but it is also lazy, unscientific and wrong.

East Asian countries have the highest average test scores, but also some of the most uneven results. The gap between their best and worst pupils is larger than anywhere else. Should British teachers try to increase the gap between the best- and worst-performing students, in the hope of dragging up the average? No. There is no evidence that the inequality of outcomes in east Asian schools contributes to raising standards. The west should not copy its rivals’ methods before it knows which ones are responsible for their success.

One policy that can increase general standards of accomplishment while reducing inequality is injecting competition into the system. This forces poorly performing schools to improve or lose students (and funding) to providers such as academies in the UK and charter schools in the US, which enjoy a large degree of independence from the bodies that fund them. But these measures remain controversial. Much of the debate centres around whether academies perform better than other schools on the most important measures of student attainment.

These arguments miss the point. It is difficult to be sure whether one school performs better than another because it is an academy, because it is funded more generously, or because it attracts students who would have performed well wherever they went to school. Pointing to the success of a small number of UK academies is no better than copying indiscriminately from the countries that have the highest grades.

In a recent study, two education economists – Martin West of Harvard and Ludger Woessmann of the University of Munich – have broken out of this short-term perspective and presented compelling evidence that competition works. Their trick is to make clever use of an accident of history. In the 19th century the Roman Catholic church faced a crisis of influence as protestant governments set up school systems that were beyond Papal influence. The Vatican responded by setting up its own schools. This early experiment in a competitive school system has left its mark, in the form of a parallel education system that persists to this day.

Ecclesiastical influence did not operate evenly in all places. In countries such as France and Italy, where Catholicism remained a political force, there was no need for the church to compete with the state. Elsewhere, it was in countries with a sizeable Catholic minority that the Church was most successful in setting up an alternative to state education. Prof West and Prof Woessmann find that in these countries – where the legacy of the Vatican’s efforts is competition between schools – attainment is strongest. The effect remains even after adjusting for socioeconomic factors that influence grades.

Some fear that allowing schools to compete with one another will heighten inequality. These critics argue that there is no point in competition unless some schools are worse than others, and there will always be some students stuck in underperforming schools. But competition can force laggards to improve. In fact, there is evidence that competition actually decreases the impact of pupils’ backgrounds in international tests.

Western schools need to do better. But copying the country with the highest grades is not the right way to improve.