what constitutes an adequate grasp of the subject.

_b. The pupil's knowledge and his methods of study._--Entirely aside
from the question of the preparation of the lesson assigned, the
teacher must constantly test the pupil's knowledge in order that he
may know how and what next to teach him; for no maxim of teaching is
better established than that we should proceed from the known to the
related unknown. And this is only another way of saying that we should
build all new knowledge upon the foundation of knowledge already
mastered.

To illustrate: Pupils must have a thorough mastery and ready knowledge
of addition, subtraction, multiplication, and division before we can
proceed to teach them measurements or fractions. And without doubt
much time is wasted in attempting to teach these subjects without a
ready command of the fundamental operations. Further, pupils must
know well both common and decimal fractions before they can proceed to
percentage. They must know and be able to recognize readily the
different "parts of speech" before they can analyze sentences in
grammar.

But not less important than what the pupil knows is _how_ he knows the
thing; that is, what are his methods of study and learning. The pupil
in a history class may be able to recite whole pages of the text
almost verbatim, but when questioned as to the meaning of the events
and facts show very little knowledge about them. A student confessed
to her teacher that she had committed all her geometry lessons to
memory instead of reasoning them out. She could in this way satisfy a
careless teacher who did not take the trouble to inquire how the pupil
had prepared her lessons, but she knew little or no geometry.