Characteristics of difficult problems

As elucidated by Dietrich Dörner and later expanded upon by Joachim Funke, difficult problems have some typical characteristics that can be summarized as follows:

* Intransparency (lack of clarity of the situation)
o commencement opacity
o continuation opacity
* Polytely (multiple goals)
o inexpressiveness
o opposition
o transience
* Complexity (large numbers of items, interrelations, and decisions)
o enumerability
o connectivity (hierarchy relation, communication relation, allocation relation)
o heterogeneity
* Dynamics (time considerations)
o temporal constraints
o temporal sensitivity
o phase effects
o dynamic unpredictability

The resolution of difficult problems requires a direct attack on each of these characteristics that are encountered.

In reform mathematics, greater emphasis is placed on problem solving relative to basic skills, where basic operations can be done with calculators. However some "problems" may actually have standard solutions taught in higher grades. For example, kindergarteners could be asked how many fingers are there on all the gloves of 3 children, which can be solved with multiplicatio