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Even fairly good
students, when they
have obtained the
solution of the
problem and written
down neatly the
argument, shut their
books and look for
something else.
Doing so, they miss
an important and
instructive phase of
the work. ... A good
teacher should
understand and
impress on his
students the view
that no problem
whatever is
completely
exhausted.
One
of the first and
foremost duties of
the teacher is not
to give his students
the impression that
mathematical
problems have little
connection with each
other, and no
connection at all
with anything else.
We have a natural
opportunity to
investigate the
connections of a
problem when looking
back at its
solution.
How to Solve It. Princeton: Princeton University Press. 1945.
Loci: Convergence
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Mathematical Treasures
Zhoubi suanjing
These two pages are from the Zhoubi suanjing (Arithmetical Classic of the Gnomon and the Circular Paths of Heaven), a Chinese book on astronomy and mathematics dated to approximately 100 BCE. These images are from a Ming dynasty copy printed in 1603. These diagrams were added to the original text at some point in an attempt to illustrate a dissection proof of the "Pythagorean Theorem", known by the Chinese as the Gougu theorem. A complete English translation and analysis of the Zhoubi suanjing is given by Christopher Cullen in his Astronomy and mathematics in Ancient China: the Zhou bi suan jing (Cambridge University Press, 1995). See, in particular, appendix one.
On this page, the diagram on the right is usually called the "hypotenuse diagram" and illustrates the proof of the theorem in the 3-4-5 case. The diagram on the left shows how a square of side 3 fits into a square of side 5.
This diagram illustrates a square of side 4 fitting into a square of side 5.