Dramatic variations in pupils’ ability to manipulate numbers have been uncovered by a Government survey of maths teaching across three urban authorities.

In one school, only 4 per cent of six and seven-year-olds were able to write down a two-digit number when it was expressed in words, while in another, 93 per cent were able to do so.

The report, published this week by the Office for Standards in Education, suggests that teachers do not pay enough attention to pupils’ ability to memorise number facts. Weaknesses in the teaching of mental arithmetic were found in many of the 45 schools visited. The availability of calculators meant that teachers no longer expected pupils to learn tables and number facts by heart.

The investigation focused on Year 2 (age six and seven) and Year 6 (age 10 and 11) in Greenwich and Newham in London and Knowsley in Merseyside, using tests devised by Leeds University combined with lesson observation.

The seven-year-olds achieved results broadly in line with national norms, but the older children tended to perform less well, with girls’ performance in particular falling behind as they grew older.

The relationship between social factors and performance was also more marked in the older group. Mental arithmetic was given too little attention in both groups, the report says.

It did, however, emphasise that excellent teaching was found in many of the schools, and identified characteristics common to good and bad lessons. The successful lessons always included a balance of whole-class, group, and individual work, with the best lessons including the most whole-class teaching, with teachers offering clear explanations and asking searching questions.

Bad lessons suffered from a “debilitating over-use of individual work and, to a lesser extent, group work”. Pupils received little direct teaching and struggled with worksheets on their own.

The report admits that all three boroughs have high levels of poverty and pupils with English as a second language. It is notably less negative in tone than its controversial predecessor on reading in 45 inner-city primaries.

After the publication of the reading report last year, the three London boroughs involved protested that negative commments had been inserted at the last minute.

The reading report was also used by the Government to justify the reform of teacher training. Before agreeing to co-operate in the numeracy study, Greenwich, Knowsley and Newham sought assurances as to how the study would be conducted.