According to the legend, he jumped out of the tub and ran from the bathhouse naked screaming, “Eureka, eureka! have the same root in the ancient Greek language and so it has been claimed that this is how the academic discipline of “heuristics” dealing with effective approaches to problem solving (so-called heurisms) was given its name. Its specific goals include highlighting in general terms the reasons for selecting those moments in a problem the examination of which could help us find a solution. 5) This discipline has grown, in part, from examining the approaches to certain problems more in detail and comparing them with each other in order to abstract similarities in approach, or so-called heurisms.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.In the survey, four interrelated areas are reviewed: (i) the relevance of heuristics in problem solving approaches—why are they important and what research tells us about their use?(ii) the need to characterize and foster creative problem solving approaches—what type of heuristics helps learners think of and practice creative solutions?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.To this end, we have assembled four summaries looking at four distinct, yet inter-related, dimensions of mathematical problem solving.The first summary, by Regina Bruder, is a nuanced look at heuristics for problem solving.

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