This brief paper welcomes the notion of local pockets of resistance while raising two problems about it. One of these concerns teachers' values and leads into wider considerations about what the underlying aims of education should be. This prepares the ground for the 'deeper pocket' of the title. This has to do with challenging the lack of official -- i.e. public, not hidden -- aims informing English school curricula. In this, England will soon be at odds with the three other UK nations, all of which, as from September 2022, will have aims-based curricula. A many-sided national campaign is recommended to target this weakness in the English system, working closely with more local pockets of resistance.