Too many schools are "teaching to the test" in mathematics, stifling genuinely stimulating thinking about the subject, a report suggests.
Students must learn to reason, Ofsted said
Education watchdog Ofsted looked at 26 schools, sixth forms and other colleges in England and found that about half of lessons failed in this regard.
Many 14 to 19-year-olds "did not expect to understand mathematics", it said.
But a government spokesman said GCSEs and A-levels could not be passed without "mastering" the subject.
'Incorrect, incomplete, inappropriate'
Ofsted found many lessons "lacked sufficient flair, imagination and challenge to get the best from students".
They did not allow them to develop their "ability to reason and discover solutions for themselves".
In a few GCSE lessons, they were given "incorrect, incomplete, inappropriate or misleading information".
Ofsted recommended that teaching at this stage should focus on "high levels of performance and secure understanding" to prepare students for going on to A-level.
A shortage of qualified maths teachers left some groups taught by non-specialists.
In one further education college, a group working on the "application of number" was taught by a tutor whose highest maths qualification was a grade D at GCSE.
Ofsted's director of education, Miriam Rosen, said: "At present too many students do not expect to understand mathematics.
"Students try to pass exams by memorising lots of unconnected facts rather than a few guiding principles.
"The current approach to teaching mathematics is not giving students the understanding they require and this must change."
Schools minister Jim Knight said: "We do not accept that tests can be passed without properly mastering and understanding the subject and preparing pupils to demonstrate mastery of the curriculum does not mean 'teaching to the test'."
The Association of School and College Leaders said Ofsted's criticism of schools and colleges for teaching to the test in maths was amazingly hypocritical.
General secretary John Dunford said: "Schools are judged by Ofsted on the results of those tests, and head teachers are losing their jobs when Ofsted decides that pupils are not doing well enough in the tests.
"Instead of criticising the teaching, Ofsted should be criticising the tests, on which schools not unreasonably base their classroom work."
The Ofsted report follows a government-commissioned review of maths in 2004 by Professor Adrian Smith, principal of Queen Mary, University of London.
He said maths was so central to much of the modern economy it should be treated as special, not on a par with other subjects.
He suggested further cash incentives to attract maths teachers and to encourage pupils to keep studying it.
On the other hand, David Burghes, from Exeter University, argues that most children learn all the maths they need by the age of 14 or even 11.
So letting children who were not interested in the subject drop it would mean better teaching for others.
Prof Smith said the latest Ofsted report echoed his own findings. A key problem remained the shortage of qualified, confident maths teachers.
He told BBC News the government had taken steps to address this - he had regular meeting with Schools Minister Andrew Adonis and "I can't fault the commitment that we have seen".
But recruiting more teachers and improving teachers' skills and confidence took time.
League table pressure
There was "a rock and a hard place" in the government's decision to require higher-grade maths and English GCSEs as the new benchmark for school league tables.
"I think it's important to send a signal that maths and English should be at the core," he said.
"If you have got someone who is less qualified and less confident, given the climate of league tables and given the pressures within the school, they are likely to prioritise teaching to the test."
But this was nothing new - his own report had identified the problem.
"So I don't think there's an argument that says the government should reconsider that," he added.