Once the teacher gives up the search for precision, the students will as well.
Once the textbook writer gives up the search for precision, the pre-service teacher will as well. As will the college professor.
The correct concept, concisely and precisely stated, is an ideal not often reached even in math. As we go more in that direction, we tend to leave out details and completeness suffers. This is indicated by people not getting it.
The epsilon delta definition in calculus is a case where a precise set of words for the parts of the definition were abandoned as a community goal. Such terms as output and input tolerance, base point and test point, and the like may exist here or there, but seldom in combination. The series Base Point Test Point Calculus is calculus in words using such terms and their combinations with full exactness and without parts or steps left out.
It parallels standard calculus and so at times becomes prolix.
Vanishing Calculus in words is the next iteration in this search. it stays in words but finds new words and phrases to make the longer approaches of Base Point Test Point Calculus go away without sacrificing rigor or completeness.
The result is to get closer to Lennes Veblen, Cauchy and Lagrange’s program for calculus.
He was a high school teacher during part of his career. The Lennes Veblen book is a rejection of the pit of self discovery. It is clarity against the tide of math man’s incomprehensibility to students and readers of math.
Base Point Test Point Calculus and Peano Axioms books at e-vendor.