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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.Mental activity starts with a goal and the motive of a person to perform such activity.
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.
The need of making students aware of heuristic approaches is by now largely accepted in didactic discussions.
Differences in varying approaches to promoting problem-solving abilities rather refer to deciding which problem-solving strategies or heuristics are to imparted to students and in which way, and not whether these should be imparted at all or not.).
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.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.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.In 1983 Zimmermann provided an overview of heuristic approaches and tools in American literature which also offered suggestions for mathematics classes.In the German-speaking countries, an approach has established itself, going back to Sewerin (), in a first phase of research on problem solving, particularly in the 1960s and the 1970s, a series of studies on problem-solving processes placing emphasis on the importance of heuristic strategies (heurisms) in problem solving has been carried out.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.It has infused mathematics curricula around the world with calls for the teaching problem solving.And as such, it has been of interest to mathematics education researchers for as long as our field has existed.