You could send for a birth certificate of one of the two children i.e. Gordon Stanley GARDNER, born in 1906 Newport M and Providence Annie GARDNER, born 1909 Newport M.

If you use the website FreeBDM - and search for these two births, this would give you the information. I think there are instructions on FOD on how to go about sending for certificates from the General Records Office - which also has a website - http://www.gro.gov.uk/gro.contents

The birth certificates would give the maiden name of the mother - and if Beatrice Harriet did marry a LEWIS in Newport M (see previous reply) prior to marrying Herbert GARDNER, this surname would possibly be on the children's certificate as well, as was the case with my Grandmother's birth certificate.