Progress and achievement.  Have you made any progress?
What was the essential achievement? We may address
dress questions of this kind to ourselves when we are
solving a problem or to a student whose work we super
supervise. Thus, we are used to judge, ore or less confidently,
progress and achievement in concrete cases. The step
from such concrete cases to a general description is not
easy at all. Yet we have to undertake this step if we wish
to make our study of heuristic somewhat complete and
we must try to clarify what constitutes in general, progress
and achievement in solving problems.
1.    In order to solve a problem, we must have some
knowledge of the subject-matter and we must select and
collect the relevant items of our existing but initially
dormant knowledge. There is much more in our 
conception of the problem at the end than was in it at the
outset; what has been added?  W Hat we have succeeded
in extracting from our memory.  In order wot obtain the
solution we have to recall various essential facts. We have
to recollect formerly solved problems, known theorems
definitions, if our problems is mathematical. Extracting
such relevant elements from our  memory may be termed 
mobilisation.
2.     In order to solve a problem, however, it is not 
enough to recollect isolated facts, we must combine these
facts, a d their combination must be well adapted to the
problem at hand. Thus, in solving a mathematical problems-
,, we have to construct an argument connecting the
materials recollected to a well adapted whole. This
adapting and combining activity may be termed organisation/
    3. In fact mobilisation and organisation can never be
really separated.  Working at the problem with the concentration,
we recall only facts which are more or less
connected with our purpose, and we have nothing to connecting
and organise but materials we have recollected and
mobilised.
      Mobilisation and organisation are but two aspects of
thethe same complex process which has still many other 
aspects.
     4. another aspect of the progress of our work is that
our mode of conception changes Enriched with all the ,
materials which we have recalled, adapted to it, and worked
into it, our conception of the problem is much
fuller at the end than it was at the outset. Desiring 
tao proceed from our initial conception of the problem to a 
more adequate, better adapted conception, we try various 
standpoints and view the problem from different sides.
We could make hardly any progress without VARIATION OF THE PROBLEM
5.     As we progress towards our final goal we see more
and more o it, and when we see it better we judge that we
are nearer to it. As our examination of the problem ad-
advances, we foresee more and more clearly what should
be done for the solution and how it should be done. Solving
a mathematical problem we may foresee, if we are are lucky
that a certain known theorem might be used, that the
consideration o a of a certain formerly solved problem might
be helpful, that going back to the meaning of a certain
technical term might be necessary.; We do not 
foresee such things with certainty, only with a certain degree of
plausibility. We shall attain complete certainty when we
have obtained the complete solution, but before observing
certainly we must often be satisfied with a more or
less plausible guess. Without considerations which are
only plausible and provisions we could never find the 
solution which is certain and finals. We need HEURISTIC REASONING
     6.  What is progress towards solution? Advancing
mobilisation and organisation of our knowledge, evolution
of our conception of the problem, increasing prevision
of the steps which will constitute the final
argument. We may advance steadily, by a small imperceptible
steps, but now and then we advance abruptly
by leaps and bounds. A sudden advance toward the solution
is called a BRIGHT IDEA, a good idea, or happy 
thought, a brain-wave(in German there is a more technical term,
Einfall). What is a bright idea? An abrupt
and momentous change of our outlook, a sudden re-organisation
of our mode of conceiving the problem, a just
emerging confident prevision of steps that we have to take
in order to attain the solution.
7.     The foregoing considerations provide the questions 
and suggestions of our list with the right sort of back-
ground.
    Many of these questions and suggestions aim directly at
the mobilisationof our formerly acquired knowledge
Have you seen it before? Or have you seen the same prom-
hem in a slightly different form? Do you know a related
problem? Do you know a theorem that could be useful?
Look at the unknown! And try to think of a familiar
problem having the same or a similar unknown.
      There are typical situations in which we think that we 
have colleected the right sort of material and we work for 
a betterorganisation of what we have mobilised: Here
is a problem related to yours and solved before. Could
you use it? Could you use its result? Could you use its
method? Could you introduce some auxiliary element in
order to make its use possible?
    There are other typical situations in which we think 
that we have not yet collected enough material. WE
wonder what is missing: Did you use all the data? Did 
you use the whole condition? Have you taken into account-
count all essential notions involved in the he problem?
     Some questions aim directly at the variationof the 
problem: Could you restate the problem? Could you
restate it still differently? Many questions aim at the 
variation of the problem by specific means, as going
back to DEFINITION,using ANALOGY,< a href="generalisation.html">Generalisation,jSPECIALISATION,DECOMPOSING AND RECOMBINING
     Still other questions suggest a trial to foresee the nature-
 of the solution we are striving to obtain: Is it possible-
l to satisfy the condition? is the condition sufficient
to determine the unknown? Or is it in sufficient? Or
redundant? Or contradictory?
     The  Questions and suggestion of our list do not men
mention directly the  bright idea; but in fact, all are con-
concerned with it. Understanding the problem we prepare
for it, devising a plan we try to provoke it, having pro-
provoked it we carry it through, looking back at the course
and the result of the solution we try to exploit it better8(footnote: Several points discussed in this article are more fully considered
in the author's paper, Acta Psychological, Vol 4, 1938, pp. 113-170).