Do symbols and graphs let people pretend they understand?

Is the popularity of the symbolic in math ed that it lets everyone pretend something is understood?  We draw a graph and then we can move on as if the students understood everything that makes the graph work and convey the result.

The best example is Euler in 1765 saying the line justifies fractions and then proceeding to state the rules of fractions without the slightest attempt to prove them from “line facts”.

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About New Math Done Right

Author of Pre-Algebra New Math Done Right Peano Axioms. A below college level self study book on the Peano Axioms and proofs of the associative and commutative laws of addition. President of Mathematical Finance Company. Provides economic scenario generators to financial institutions.
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