Teachers use self-discovery and make their students be their teachers. Their students don’t know and don’t self-discover the Peano Axioms. The result is the teachers don’t know they don’t understand elementary math. Or can even imagine what the math of elementary math looks like or sounds like.
When confronted with it as in the Devlin “It ain’t repeated addition”, they throw up their hands, say Peano Axioms are too hard and go back to self-discovery. They don’t see the contradiction.
If they can’t understand Peano Axioms, then self-discovery isn’t working. They have not self discovered the concepts of elementary math because when confronted with Peano Axioms their response is it is impossible for them to understand the math of elementary math.
The reaction is that teachers and students could never understand the Peano Axioms or teach them. This means the whole self discovery movement admits it could never discover elementary math, nor the conceptual meaning of elementary math or explain it to others.
Then they promptly ignore this and go back to thinking they know elementary math as they use self discovery with students who don’t expose the flaws and gaps in their thinking. Self discovery is a self perpetuating regime of gaps in understanding. The teachers are constantly reinforced that they know elementary math by the failure of their students to challenge the gaps in their understanding of elementary math.
This failure to find their gaps in understanding shows that self discovery does not work to find the gaps. This shows self discovery is not the concept discovery and learning machine that is claimed for it. Self discovery does not find the Peano Axioms nor analyze place value as recursive.
Self-discovery does not discover the limit definition in terms of epsilon and delta. It does not discover good explanations in words such as base point test point for the limit definition.
If self discovery was so good for concepts, then text materials would get better in stating and teaching concepts to keep up. Instead, the opposite happens. The no text or sloppy explanations of Khan Academy spread.
Base Point Test Point Calculus and Peano Axioms books at e-vendor.