The right to an education is written into state constitutions (Old Dad, remember that the federal government doesn't have the last word). For example,
the California Constitution is pretty specific:
SEC. 5. The Legislature shall provide for a system of common schools by which a free school shall be kept up and supported in each district at least six months in every year, after the first year in which a school has been established.
Yes, but even at a federal level, a right to education can be established without amending the Constitution. All that's necessary is for sufficient legislation and public consciousness to decide that a right to an education is a thing.
AFAIK, the last time the Supreme Court directly addressed this question was 1973, in San Antonio Independent School District v. Rodriguez. Since that time, we've had significant Congressional legislation (ADA, IDEA, NCLB) which clearly establish access to an education as a protected right (hello, FAPE!).
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