The study involved independent testing of more than 12,000 pupils aged from five to 14.
The government said its test results in maths had shown clear improvement since 1997 at all ages.
Professor Williams, from the School of Education, said year on year improvements in mathematics were almost nonexistent for higher and lower achievers aged between 11 and 14.
He said the study, conducted in England in 2005, showed "short term" increases in children's maths skills as they were taking national tests, but that their abilities declined again once the "Sats" were over.
"We did record short-term improvements in test scores around the period of national testing at Key Stage 1 and 2 (years two and six of primary school)," he said.
"However, these increases were short term and the overall trend continued shortly afterwards in years three and seven, suggesting lack of lasting gain in children's understanding."
