Vicky Colbert, fresh from a graduate school of education in the 1970s, faced what seemed an impossible challenge: to teach 60 children aged 5-14 in a one-room school in rural Colombia - all by herself.

Colbert soon realised that teaching such a large and varied group in the traditional way - standing at the front instructing from the blackboard - would not work. So she devised an alternative, known as Escuela Nueva (or “New School”). This method provides children with written learning guides so they can work through a subject together, in small, self-governing groups, with only occasional direct guidance from a teacher.

Since Colbert got started in 1975, Escuela Nueva has, according to the International Finance Corporation, spread to more than 27,000 Colombian schools and been taken up in 16 countries, mainly in Latin America, reaching 5 million children.

Escuela Nueva is a prime example of high-impact innovation in education. It has not just helped to get more children into school; it has also, by adopting a different kind of pedagogy, produced better results. More, different and better go hand in hand. For my book Innovation in Education, I spent much of last year researching the work of Colbert and 15 other such high-impact innovators, most of whom have won awards for their efforts from the Qatar Foundation’s World Innovation Summit for Education (WISE).

Education is awash with innovation, but much of it remains small-scale and incremental. What stood out about the 16 groups I researched was that they were all challenging the traditional school model, at scale. In the case of The Citizens’ Foundation in Pakistan, a WISE award-winner in 2010, this means reaching more than 100,000 boys and girls with a network of almost 800 community-funded schools that employ only women teachers.

Elsewhere, from a tiny base in Paraguay, 2009 WISE award-winner Martin Burt’s Self-Sufficient School has launched a worldwide movement of self-financing farm schools: the children help to grow the produce that funds the school. At the other end of the scale, the Massachusetts Institute of Technology (MIT) - a 2010 WISE award-winner - produces OpenCourseWare materials that have been visited more than 100 million times, much of that traffic coming from independent learners.

These high-impact innovators provide many lessons for policymakers in the developing and developed worlds: innovation often starts in marginal places; the innovators often draw on cosmopolitan sources; they seek to increase access to education, but also to make it more motivating, engaging and useful; they grow and exert influence by operating like social networks and movements; they tend to be frugal.

However, the most important lesson is about the conditions that make learning come alive. This lesson has been drawn from organisations such as the mothers’ groups run by the Mother Child Education Foundation in Turkey and the Next Einstein Initiative postgraduate mathematics seminars at the African Institute for Mathematical Sciences near Cape Town in South Africa (both WISE winners in 2010), as well as the community-based learning run in favelas in Sao Paulo, Brazil, by Aprendiz City-School.

These programmes find the right mix between two ingredients. The first is that learning has to have structure: an underlying skeleton of knowledge, capabilities and skills to be acquired. The second is that all these programmes are highly empathic: the people involved learn through open, sometimes intense relationships, both between teachers and pupils and among the pupils themselves. It is these relationships that make learning engaging, exciting and open-ended.

The best places to learn, whether in families, workplaces or schools, have strong underlying systems that support but do not dominate or displace intense relationships. The best schools the world over, from the charter chain High Tech High in San Diego, California, and the state Kirkkojarvi school in Espoo, Finland, to the UK’s private-sector Brighton College, have this recipe at their heart. They have robust systems - for keeping track of achievement, for example - but combine that with highly intense and supportive relationships.

But the warm glow of empathic relationships alone is not enough. The fun teacher who never gives honest feedback is a patronising fake. Learners want a sense of purpose and challenge, to be stretched as well as made to feel good.

The worst place to be is the kind of secondary school I went to, of which there are still far too many, where there are neither systems nor empathy, just hundreds of children, a demoralised workforce and barely contained chaos.

Faced with chaos, a bit of discipline and method is of course welcome. That is the inspiration behind the highly successful Knowledge Is Power Program (KIPP) charter chain in the US. But schools that become over-structured, which teach by routine and to the test, all too easily become busy but dead, systematised but soulless. Waves of reform all around the world threaten to containerise education, delivering children to their destinations as if they were packages from Amazon.

Simply getting more children to sit still in class for longer listening to a teacher, even a good one, is not a recipe for learning. That is Tove Wang’s conclusion. And she should know, because in 2005, with a handful of colleagues at Save the Children, she launched an inspirational programme, Rewrite the Future. In five years, it got 1.8 million children in conflict zones into school for the first time and improved schooling for almost 10 million children. At the end of it, Wang concluded that they had chosen the wrong goal: they got children into school, only for many of them to be subjected to hours of rote learning.

More school is not enough. We need different and so better ways to learn. The key to that is to create, inside and outside school, systems and structures that allow for empathic and strong relationships. That is the innovator’s secret.

Charles Leadbeater is author of Innovation in Education: Lessons from Pioneers Around the World, supported by the World Innovation Summit for Education and published by Bloomsbury.