Opinions about mathematicians differ widely. Some think they are gifted, others that they are untalented – which is why mathematicians themselves prefer to have nothing to do with it and at worst even allow themselves to be branded as unintelligent. However, recent studies show that in mathematics, as in other disciplines, talent or and intelligence are not the only factors at play.

In her lecture “Intelligent knowledge as the key to ability”, given at ETH Zurich before Christmas, ETH Zurich Professor Elsbeth Stern said that although intelligence is as important as ever and the IQ test is still a suitable instrument to determine a person’s intellectual potential, innate intelligence plays a greater role in countries where there is widespread equality of opportunity. Stern has found from various studies that “The influence of genes even increases with age.” However, when background knowledge comes into play, differences in intelligence lose importance.

Knowledge and IQ act in combination

Stern says "Just being intelligent without having background knowledge is of no use.” In other words, no matter how intelligent a person is, if her background knowledge is weak she will be inferior to another less intelligent person if the latter has the appropriate (background) knowledge. The intellectual game of chess is exactly where this becomes apparent. An intelligent novice is most unlikely to beat an experienced but less intelligent player. Wide-ranging studies of the learning behaviour of school students in Bavaria also showed that mathematics is the very subject where intelligence is not the only thing that affects performance. Pupils who were not already good in mathematics in Year 2 did not become so in Year 11 either. However, the way to use mathematics intelligently has more to do with knowledge than with intelligence. The students had accumulated knowledge deficits that they were scarcely able to make up for again afterwards.

Latin scholars are no better at maths

Transfer effects of what has been learnt into new content areas are also only rarely observed. A successful chess player does not become a successful military strategist and a good Latin scholar does not become an equally good mathematician merely because Latin promotes a certain language logic. According to Stern, transfer effects are difficult to achieve because people learn in specific areas rather than in an abstract formal way. However, the brain – which has remained practically unaltered for the last 40,000 years – is organised for analogue thinking. A person who encountered something in the past will re-use it to solve a new task. This is why Stern also found that transfer effects from French to Spanish were larger than from Latin to Spanish.

However, some education-minded politicians and teaching professionals still hold a populist view of learning, as if to say that mind training and learning how to learn are sufficient to make students competent. However, according to the ETH Zurich professor, competence means possessing widely applicable knowledge that can be brought into action to cope with a class of demands, i.e. having intelligent knowledge. That is why the aim of education must be competence.

ETH Professor Elsbeth Stern: “Pupils who were not already good in mathematics in Year 2 will not become so in Year 11 either.” large

„The educational psychologist adds that “The only way to become competent is through dealing with demanding contents.” She says more use must be made of this approach in schools. From nursery school to university, students must be confronted with tasks that they cannot solve immediately but for which in principle they already have the knowledge prerequisites.

To promote the interplay between knowledge and intelligence in mathematics, challenging problem exercises must be set. Here, Stern detected a shortcoming in the German educational system. She pointed out that demanding problem exercises occur more than ten times as often in school textbooks in Eastern European countries than in the books in the German Federal Republic.

The right mixture of learning methods does the trick

What does educationally effective teaching look like? “Move away from the method-based approach,” is her demand. The learning environment must be arranged around the students’ knowledge. The scientist says that “Good teachers use various methods in their teaching, including face-to-face teaching.” She says that autonomous exploratory learning may be a problem for some students under certain conditions. “Methods must be matched to pre-existing knowledge.” Teachers must work on knowledge to make it flexible. Stern also drew attention to the importance of the teaching staff’s professional knowledge, and pleaded for subject-specific teaching staff to be employed even at primary school level. She said that teachers themselves must have a large amount of expert knowledge to enable them to formulate intelligent questions and exercises. In addition, a good teacher would set tasks by which the students could try something out for themselves. However, she said that the danger of this kind of activity-oriented teaching was that it resulted in a “hands on, minds off” situation because students did not really know what they were doing. This was why, particularly in open learning situations, teachers were indispensable in encouraging students in activities that create meaning.

Looking into students’ heads

Above all, Elsbeth Stern placed great value on the teaching staff: “A good teacher knows what is going on in the students’ heads.” He or she must understand the knowledge already present and must detect mistaken ideas. A good teacher must also know the effects of the presentation of information, must provide the students with a large repertoire of cognitively activating tasks and must recognise opportunities to encourage interdisciplinary competencies. She added: : “That calls for systematic observation of the students in their learning behaviour.“