*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.*

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.).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.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.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.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.(iii) the importance for learners to formulate and pursue their own problems; and (iv) the role played by the use of both multiple purpose and ad hoc mathematical action types of technologies in problem solving activities—what ways of reasoning do learners construct when they rely on the use of digital technologies and how technology and technology approaches can be reconciled?Mathematical problem solving has long been seen as an important aspect of mathematics, the teaching of mathematics, and the learning of mathematics.

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