They are children who adopt an extensive repertoire of tactics to avoid working in lessons, especially on tasks which may show them up and reveal poor performance.
Outside school, they may participate in the "slow riot" - petty vandalism or crime, abuse of alcohol or other drugs, or they may hang around in menacing gangs on the street corner. Some disappear from school life completely.
Two-thirds of the children excluded from primary schools are known to social services departments and three-quarters of all children who are excluded are below average intelligence.
In secondary schools excluded children on average have a reading age of between 8.5 and 10 years.
Coincidentally, research conducted for Judge Stephen Tumim, the chief inspector of prisons, discovered that three-quarters of prisoners on remand had reading ages under 10.
HMI's survey of exclusions last year discovered that there had been 11, 084 permanent exclusions in 1994-95. With eight million pupils in school, that worked out at 0.13 per cent.
Research by Carl Parsons, of the University of Kent at Canterbury, based on information gathered from 92 local authorities, put the figure higher at 13,400 - well under 1 per cent (0.17 per cent).
That means the school system maintains around 99.8 per cent of all pupils and David Moore, the HMI responsible nationally for behaviour and discipline, said: "That is a success story that we are not celebrating."
Inevitably, however, it is the disruptive children who hit the headlines and there is deep division between the teacher trade unions on the best way to tackle the problem.
The National Association of Schoolmasters Union of Women Teachers puts the interests of the teachers first and believes that disruptive children should be expelled.
The National Union of Teachers said damaged children should be helped by schools and not "demonised".
Bad children and criticism of the education system are not a new phenomenon. As far back as 1885, a Government commission reported that the German education system was better than ours.
In 1912 The Times said that most people, even the educated ones, could not spell. It claimed parents no longer read to their children, who in turn spent too much time listening to the gramophone.
But Mr Moore believes that today's youngsters have not learned the art of contrition. "When the teacher challenges them, they stare the teacher out. "
Children of families under financial or emotional stressare more likely to engage in behaviour leading to exclusion, as are pupils with low levels of literacy.
The most common reasons for exclusions are: * verbal abuse to staff; * violence to other pupils; * persistently breaking school rules; * disruption; * criminal offences, usually theft or substance abuse.
The socio-economic context of a school is sometimes associated with rates of exclusion, but a high number of children on free school meals - the most commonly-used indicator of poverty - does not necessarily spell trouble.
Mr Moore said there were primary schools with no free school meals where there had been 13 incidents provoking exclusions, and primary schools with 99 children on free school meals with none.
In one local authority, at least two-thirds of the primary exclusions came from three or four schools.