It wants key stage 1 tests to be scrapped and pupils to be discouraged from taking GCSE early, to allow more time for teachers to deepen understanding.
In a submission to the Government's inquiry into the future of the subject, the association welcomes the key stage 3 strategy for achieving its aim, in many cases, of improving teaching in the early secondary years.
But, it claims, this success is being undermined by the constant pressure to get what seem to be good results in league tables, results which actually reflect "inadequate" levels of understanding among pupils.
"Important aspects of mathematical learning that are hard to assess become optional in the eyes of students and teachers," said the association's evidence.
Doug French, chairman of the association's teaching committee, said that schools would increasingly drill pupils on set ways of solving problems, rather than giving them the tools to approach the questions with a deeper understanding.
He said that, as a result, universities were complaining that undergraduates lacked sufficient knowledge of maths, a concern echoed by A-level teachers.
The association says key stage 2 and 3 tests, GCSEs and A-levels should stay, but wants an end to optional testing and AS-levels in the first year of sixth form.
It says that maths teaching has been blighted by teacher shortages and needs to be greatly improved. Teachers should be offered "substantial" professional development work to boost their subject knowledge.
The evidence is the latest in a series of attacks on testing and targets in schools, coming a week after chief inspector David Bell said that national targets were demotivating teachers.
The maths inquiry, headed by Professor Adrian Smith, principal of Queen Mary College, London, is due to report in the summer.