Can you check the result?   Can you check the argument?
A good answer to these questions strengthens our
trust in the solution and contributes to the solidity of our
knowledge.
1.   Numerical results of mathematical problems can be 
tested by comparing them to observed numbers, or to a 
commons sense estimate of observable numbers. As problem
 arising form the practical needs or the natural curiosity 
almost always aim at facts it could be expected that such
comparisons with observable facts are seldom omitted.
Yet every teacher knows that students achieve incredible 
things in this respects. Some students are not disturbed
at all when they find 16, 130 ft. for the length of the boat
and 8 years, 2 months for the age of the Capitan who is
by the way, known to be a grandfather. Such neglect of
the obvious does not show necessarily stupidity but rather
indifference towards artificial problems.
2    Problems "in letters" are susceptible of more, and 
more interesting, tests than "problems in numbers" (section 14). 
For another example, let us consider the frustum of a pyramid with square base. If the side of 
the lower base is a, the side of the upper base b, and 
the altitude of the frustum h, we may find for the volume
((a^2+ab+b^2)/3)*h

We may test this result by SPECIALISATION.
In fact, if b=a the frustum becomes a prism and the formula
yields a^2h and if b0 the frustum becomes a pyramid
and the formula yields a^2h/3. We may apply the 
TEST BY DIMENSION. 
In fact, the expression has as dimension the 
cube of a length. Again, we may test the formula by
variation of the data. In fact, if anyone one of the positive
quantities a,b, or h increases the value of the expression
increases.
    Tests of this sort can be applied not only to the final
result but also to intermediate results. They are so useful 
that it is worth while preparing for them. See 
VARIATION OF THE PROBLEM
In order top be able to use such tests,
we may find advantage in generalising in "problem in 
numbers and changing it into a problem in letters see
GENERALISATION 
3. Can you check the argument?  Checking the argument
step by step, we should avoid mere repetition. First
mere repetition is apt to become boring, uninstructing, a
strain on the attention. Second, where we stumbled once, 
there we are likely to stumble again if the circumstances
are the same as before. If we feel that it is necessary to go 
again through the whole argument step by step, we should
at least change the order of the steps, or their grouping, to 
introduce some variation.
4.    It requires less exertion and is more interesting to 
pick out the weakest point of the argument and examine
it first. A question very useful in picking out points of
the argument that are worth while examining is:
DID YOU USE ALL THE DATA?
5.   It is clear that our non-mathematical knowledge can-
not be based entirely on formal proofs. The more solid part
of our every day knowledge is continually tested an
strengthened by our everyday experience. Tests by observation
are more systematically conducted in the natural sciences whose 
result is successfully tested.  Observations are more 
systematically conducted in the natural sciences. Such tests take
the form of careful experiments and measurements, and are combined
with mathematical reasoning in the physical sciences. Can our
knowledge in mathematics be based on formal proofs
alone?
      This is a philosophical question which we cannot debate here.
it is certain that your knowledge, or my knowledge, or your students,
knowledge in mathematics is not based on formal proofs alone. 
If there is any solid knowledge at all it has a broad experimental basis,
and this basis is broadened by each.