*What is interesting, though, is the diverse ways in which each of the four aforementioned contributions draw on, and position, these works so as to fit into the larger scheme of their respective summaries.This speaks to not only the depth and breadth of these influential works, but also the diversity with which they can be interpreted and utilized in extending our thinking about problem solving.*

Taken together, what follows is a topical survey of ideas representing the diversity of views and tensions inherent in a field of research that is both dates back to the time of Archimedes and is said to have come out of one of the famous stories told about this great mathematician and inventor.

The King of Syracuse asked Archimedes to check whether his new wreath was really made of pure gold.

This is why they are also often unable to explain how they actually solved a given problem.

To be able to solve problems successfully, a certain mental agility is thus required.

Archimedes struggled with this task and it was not until he was at the bathhouse that he came up with the solution.

As he entered the tub he noticed that he had displaced a certain amount of water.More relevant, mathematical problem solving has played a part in every ICME conference, from 1969 until the forthcoming meeting in Hamburg, wherein mathematical problem solving will reside most centrally within the work of Topic Study 19: Problem Solving in Mathematics Education.This booklet is being published on the occasion of this Topic Study Group.This notion of heuristics is carried into Peter Liljedahl’s summary, which looks specifically at a progression of heuristics leading towards more and more creative aspects of problem solving.This is followed by Luz Manuel Santos Trigo’s summary introducing us to problem solving digital technologies.Whilst the content in mathematical problem-solving consists of certain concepts, connections and procedures, the process describes the psychological processes that occur when solving a problem.This course of action is described in Lompscher by various qualities, such as systematic planning, independence, accuracy, activity and agility.The last summary, by Uldarico Malaspina Jurado, documents the rise of problem posing within the field of mathematics education in general and the problem solving literature in particular.Each of these summaries references in some critical and central fashion the works of George Pólya or Alan Schoenfeld. The seminal work of these researchers lie at the roots of mathematical problem solving.Along with differences in motivation and the availability of expertise, it appears that intuitive problem solvers possess a particularly high mental agility, at least with regard to certain contents areas. by the capacity to change more or less easily from one aspect of viewing to another one or to embed one circumstance or component into different correlations, to understand the relativity of circumstances and statements.It allows to reverse relations, to more or less easily or quickly attune to new conditions of mental activity or to simultaneously mind several objects or aspects of a given activity (Lompscher These typical manifestations of mental agility can be focused on in problem solving by mathematical means and can be related to the heurisms known from the analyses of approaches by Pólya et al. also Bruder Intuitive, that is, untrained good problem solvers, are, however, often unable to access these flexibility qualities consciously.

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