Hamilton explained what is in the power of the federal government in the Essay on the National Bank. There are implicit powers and NONE of the Founders denied that. Anything which is not forbidden explicitly and which is necessary for the carrying out of the enumerated powers and not contrary to the spirit of the Constitution is constitutional according to Hamilton. And there is no greater expert on this matter than he.
Since our national defense is heavily dependent upon an educated military there are plenty of reasons there is an federal interest in education. Technology alone demands such an involvement.
Those terms do not have to be in the document any more than the word "border" does in order to establish one.
It is a common myth around here that the Constitution was written to reduce federal power when it was written to reduce STATE power and increase federal. Almost all the limitations within the Constitution are placed upon the States.