ORIGINAL ARTICLE Problem solving in the mathematics classroom: the German perspective Kristina Reiss Æ Gu ¨nter To ¨rner Accepted: 20 May 2007 / Published online: 19 June 2007 Ó FIZ Karlsruhe 2007 Abstract In Germany, problem solving has important roots that date back at least to the beginning of the twentieth century. However, problem solving was not primarily an aspect of mathematics education but was particularly influenced by cognitive psychologists. Above all, the Gestalt psychology developed by researchers such as Ko ¨hler (Intelligenzpru ¨fungen an Anthropoiden. Verlag der Ko ¨niglichen Akademie des Wissens, Berlin, 1917; English translation: The mentality of apes. Harcourt, Brace, New York, 1925), Duncker (Zur Psychologie des produktiven Denkens. Springer, Berlin, 1935), Wertheimer (Productive thinking. Harper, New York, 1945), and Metzger (Scho ¨p- ferische Freiheit. Waldemar Kramer, Frankfurt, 1962) made extensive use of mathematical problems in order to describe their specific problem-solving theories. However, this re- search had hardly any influence on mathematics educa- tion—neither as a scientific discipline nor as a foundation for mathematics instruction. In the German mathematics classroom, problem solving, which is according to Halmos (in Am Math Mon 87:519–524, 1980) the ‘‘heart of math- ematics,’’ did not attract the interest it deserved as a genuine mathematical topic. There is some evidence that this situ- ation may change. In the past few years, nationwide stan- dards for school mathematics have been introduced in Germany. In these standards, problem solving is specifically addressed as a process-oriented standard that should be part of the mathematics classroom through all grades. This article provides an overview on problem solving in Ger- many with reference to psychology, mathematics, and mathematics education. It starts with a presentation of the historical roots but gives also insights into contemporary developments and the classroom practice. 1 Introduction Problems play a central role in the mathematics classroom, and a huge amount of learning time is designated to math- ematical problems (Reiss & Heinze, 2005; Heinze, 2007). All mathematics textbooks encompass series of problems, however working on these problems may sometimes be more or less routine for the students and can thus be quite different from real problem solving. Problem solving pre- supposes that there are a starting point and a goal, which cannot be transformed into each other by procedures immediately identified by the problem solver. Correctly calculating the sum 12,345 + 6,789 is probably a task for a student at the secondary level, which cannot be regarded as a problem but rather as an exercise. Students know the algorithms and calculation rules leading to the result and there is no barrier to be overcome. Problem solving in the sense used here (and in many other chapters of this volume) goes beyond an application of well-known rules but encompasses unknown situations, maybe unclear goals, and first of all, non-algorithmic steps that are necessary for a solution. A problem is a task the individual wants to achieve, and for which he or she does not have access to a straightforward means of solution (Schoenfeld, 1985). This chapter concentrates on problem solving in German mathematics education. We will present exemplary aspects K. Reiss (&) Department of Mathematics, Ludwig-Maximilians-Universita ¨t, Theresienstr. 39, 80333 Mu ¨nchen, Germany e-mail: kristina.reiss@math.lmu.de G. To ¨rner Department of Mathematics, University of Duisburg-Essen, Campus Duisburg, Lotharstr. 63/65, 47048 Duisburg, Germany 123 ZDM Mathematics Education (2007) 39:431–441 DOI 10.1007/s11858-007-0040-5