Problem solving in the mathematics classroom: the ... - CIMM UCR

ZDM Mathematics Education (2007) 39:431–441 DOI 10.1007/s11858-007-0040-5

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
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¨pferische Freiheit. Waldemar Kramer, Frankfurt, 1962) made extensive use of mathematical problems in order to describe their specific problem-solving theories. However, this research had hardly any influence on mathematics education—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 mathematics,’’ did not attract the interest it deserved as a genuine mathematical topic. There is some evidence that this situation may change. In the past few years, nationwide standards 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 K. Reiss (&) Department of Mathematics, Ludwig-Maximilians-Universita¨t, Theresienstr. 39, 80333 Mu¨nchen, Germany e-mail: [email protected] G. To¨rner Department of Mathematics, University of Duisburg-Essen, Campus Duisburg, Lotharstr. 63/65, 47048 Duisburg, Germany

article provides an overview on problem solving in Germany 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 mathematical 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 presupposes 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

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of the historical perspective in some detail including the psychology, the mathematics, and the mathematics education points of view. We will then discuss how German psychologists contributed to research on problem solving and why their results were significant for mathematics education. There is a small reminiscence to George Po´lya whose book on problem solving attracted some attention within the German mathematics education community. Moreover, we will address the role of problems in recent international studies on students’ competencies. Finally, the development of standards for school mathematics will be used as the basis for formulating ideas how problem solving can be integrated in the mathematics classroom.

2 Historical aspects and psychological roots of mathematical problem solving Mathematical problem solving is a fascinating task, which has attracted many researchers not only from the mathematical but also from the metacognitive perspective. Getting an idea how problem-solving works has very early been regarded as a topic of interest for researchers, both mathematicians and psychologists. In the following, we will present examples that highlight this specific interest. Moreover, we will discuss some research from psychology related to problem solving. It has to be mentioned at this point that there is an enormous amount of psychological research on problem solving. Accordingly, we will not be able to present an overview but will give a subjective choice of research related to mathematical problem solving (see, e.g., Funke, 2003, for a detailed description of psychological research). 2.1 Mathematicians and their introspection of problem-solving processes Some early reports underlining metacognitive activities have their origins in the German mathematics community. Thus Carl Friedrich Gauss’ correspondence with the astronomer Heinrich Wilhelm Olbers may serve as an early example (Fig. 1). He describes the process of finding the solution of a mathematical problem (which is not explicitly mentioned in the text) in the following way: ‘‘… this deficit put me off other results I found. During the last 4 years there was hardly a week in which I did not make the one or other futile effort in order to cut the knot.… Finally, a few days ago, I succeeded—however, not as a consequence of my cumbersome search but I dare say, by the mercy of God. Thunderstruck the puzzle unraveled; I would not be able to identify the guided fiber from what I

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Fig. 1 Letter from Gauss to Olbers, 1805, Gauss’ Werke, X, 1, p. 25

knew before and what determined my last attempts to the actual solution’’ (translation KR). This method of introspection and analyzing one’s own problem solving was continued in the nineteenth and twentieth centuries by mathematicians like Henri Poincare´, Jacques Hadamard, and Bartel L. van der Waerden. In his work on invention or discovery in mathematics, Hadamard (1949) went beyond a mere description of a single event but considered the process of invention to progress through several stages which he labeled as preparation, incubation, illumination, and verification. These stages still imply by their names the idea of a sudden inspiration but take into regard that this inspiration is embedded in a problemsolving process. According to this theory, there are stages in which the individual is absolutely conscious of the processes and others in which creative thoughts just emerge. These two aspects play a central role in the small booklet by van der Waerden (1954; partly translated into English, van der Waerden, 1983). He distinguishes (in accordance with important branches of psychology at that time) between conscious and unconscious processes during a mathematician’s work but rather chooses the concepts of inspiration and thinking in most parts of his article. The connection between both is described with the help of a small episode in which Emil Artin, Otto Schreier, and Bartel L. van der Waerden discuss a mathematical idea. The solution of the problem comes as an inspiration to van der Waerden but he is immediately sure about its truth. Obviously, inspiration guides the process of coming up with a mathematical idea, and it is an immediate process of verifying that the idea might work. However, (conscious) processes of cognition are necessary to make sure that there is a valid result. 2.2 The Gestalt psychology: Max Wertheimer and Karl Duncker It must have been reports like those mentioned in the last paragraph that attracted the attention of cognitive psy-

Problem solving in the mathematics classroom: the German perspective

chologists and led them to investigate mathematical problem solving. Moreover, the kind of mathematical problem regarded in this context had an important advantage compared to other problems as they were mostly well defined with a precise starting situation, a clear goal, and clearly specified actions that lead to a solution. The work of Max Wertheimer may serve as an example for this use of mathematical problems in psychological research. He was one of the leading Gestalt psychologists in the twentieth century, thus working within a theory, which focuses on problem solving. The Gestalt psychologists were probably the first researchers that systematically investigated processes of thinking and problem solving (cf. Opwis, 1996). A main research question was to find out what happened in a person’s mind when problem solving took place. Gestalt psychologists were particularly interested in the aspects described above: why does a person have the important idea for getting a solution of a problem? And what is this idea all about? Wertheimer (1925) claimed that the scientific psychology was only able to provide concepts like intuition but was not able to give these concepts a concrete content. His thoughts were presented to a broader public in a book titled ‘‘Productive Thinking’’ (Wertheimer, 1945; German edition 1956). In this book he discussed what distinguished productive and generative thinking from a mere reception of facts and provided a number of examples that clarified this distinction. Many of these examples were taken from (school) mathematics (e.g., identifying the area of a parallelogram, calculating the sum of the whole numbers 1,2,…,n). The chapters give their reader an idea of how problem solving can be supported in the mathematics classroom. Wertheimer (1945) describes that generating and applying a specific formula cannot be seen as a major objective of mathematics instruction. His example concerning the area of a parallelogram is specifically apt to emphasize the main idea. Wertheimer (1945) starts with describing how a teacher introduces the area of a parallelogram: he marks the vertices and drops a perpendicular from the upper right vertex to the elongated base line. Figure 2 clarifies the steps, which are verbally presented by the teacher. It is more or less a standard situation in the classroom. However, Wertheimer (1945) points out that

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Fig. 2 The area of a parallelogram

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this algorithmic approach hinders students from generalizing mathematical ideas. His distinction between analogous alterations and a mere application of presented operations is in the focus of his argumentation. Problem solving requires the first idea for successfully handling a task. Figure 3 presents an example for this distinction given by Wertheimer (1945). On the left-hand side (1) is the original task, in the middle (2) is an answer that takes the principal idea of an analogous alteration into account, on the right-hand side (3) is an answer that merely reproduces the steps of the teacher. This example suggests that it is important to make sure students understand what they are doing. Understanding a problem may institute the application of knowledge in the future that is different from repeating algorithms. According to Wertheimer (1945), productive thinking should be seen as a dynamic process. It includes aspects like looking for the inner structure of a problem, performing structuring operations, taking alterations into account. In some respect, the methods and their underlying theoretical ideas are perfect examples for a fruitful approach leading to constructive learning even though they are dating back more than 60 years. Karl Duncker was a student of Max Wertheimer despite the fact that his most influential book on productive thinking (Duncker 1935) was published 10 years before that of Wertheimer (whose book, however, appeared after his death). Duncker used mathematical problems in order to demonstrate the specific aspect of the Gestalt theory namely the restructuring of the problem. He asked his subjects, for example, why six digit numbers like 276,276, 591,591, 112,112 were divisible by 13 and offered to different groups of subjects the following hints: (a) these numbers are divisible by 1,001; (b) 1,001 is divisible by 13; (c) if a common divisor of the numbers is divisible by 13, then all numbers are divisible by 13; (d) if a divisor of a number is divisible by p, then the number is divisible by p; (e) different numbers may have a common divisor; (f) look for a subjacent commonality that suggests the divisibility by 13. A further group of problem solvers got no hint at all. The data gave evidence that hints (a) and (b) that according to Gestalt psychology aimed at the structure of the problem were most helpful (59 and 50% solution rates) whereas all other hints hardly increased the number of correct solutions (0 to 15% solution rates). Duncker (1935) underlined this result in an experiment where he presented the same task but used the numbers 276,276, 277,277, and 278,278 as

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Fig. 3 Application to a new problem

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examples. Only four subjects took part in this study, but three of them were able to solve the task without any hint. In his view, the reason might be that this series emphasizes the structural aspect as 277,277 – 276,276 = 278,278 – 277,277 = 1,001 thus leading to the divisibility by 13 of this common factor. It is noteworthy that Duncker (1935) argued in his book with mathematical topics beyond school mathematics. He provided a theory of the functional bondage of elements necessary for the solution of a mathematical problem using as an example Pasch’s axiom (‘‘a line which intersects one edge of a triangle and misses the three vertices must intersect one of the other two edges’’) and propositions based on this axiom. It is somewhat astonishing that these roots had little impact as they did neither influence the mathematics school curriculum nor, in a significant way, mathematics education as a scientific discipline. Despite the fact that conclusions for mathematics instruction might have been drawn from the experiments, they did not get the attention of the mathematics education community. A reason might be that Gestalt psychologists were not able to describe how restructuring really worked and thus had also a minor influence within cognitive psychology (Funke, 2003). However, the vicinity of some of the Gestalt ideas to results of mathematicians’ introspection is still fascinating. Both have in common that they (more or less) lack the perspective on learning and (certainly to an even greater extent) the perspective on teaching but focus on a deeper understanding of problem-solving processes and their principles. 2.3 How to solve it: George Po´lya Problem solving in German mathematics education sometimes seems to be influenced by George Po´lya and his book ‘‘How to solve it’’ (Po´lya, 1945, 1949, German translation). In this book, he suggested four steps, which might guide the solution of a mathematical problem. According to these steps, solving a problem meant (1) understanding the problem, (2) devising a plan, (3) carrying it out, and (4) looking back at the process and its results. It is worthwhile to have a closer look at the work of Po´lya because he gives some detailed hints about how to deal with mathematical problems. In particular, the four stages mentioned above are explicated in his book. The first part of the problem-solving process is to understand the problem. Specific questions can support this understanding, for example questioning what is unknown, which data are available, and what the constraints of the problem are. Moreover, there are hints, which are of a more heuristic nature. It is suggested that problem solvers draw a figure, take care of a useful notation, or identify reasonable parts. Planning the solution process is the next step, which encompasses analyzing connections between the (known

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and unknown) data, identifying auxiliary problems, changing the formulation, or checking the relevance of the data. Po´lya provides detailed information on this part of the process and gives an example from geometry. Carrying out the plan is the next step. In particular, this means to check every single part of the solution and to make sure (or, preferably, to prove) that it is correct. The last step asks the problem solver to look back and to evaluate the solution. There should be some effort to show that it is correct and all arguments are valid. Moreover, it is important to consider whether a solution could have been obtained differently or the methods used might work for another problem, as well. The book by Po´lya (1945) does not only provide this detailed instruction for problem solving but also covers an introduction on the use of heuristics in mathematical problem solving (see also Po´lya, 1950). There are distinct examples that clarify specific procedures, which play an important role in mathematical problem solving like proof techniques, backtracking, specialization, or generalization. It was probably this mixture that made the book interesting for its readers from the mathematics education and the mathematics community. However, the friendly acceptance of Po´lya’s ideas had hardly any impact on the work of scientists from both communities. In particular, there is no evidence that this approach significantly influenced classroom work in Germany. 2.4 Dynamic problem solving: Dietrich Do¨rner In Germany, problem solving as a psychological theory became once more prominent in the 1980s when Dietrich Do¨rner introduced a broadened concept, which took into account complex problem-solving processes based on illdefined problems with an uncertain result. He stated that most problem-solving research (e.g. Luchins, 1942; Klix, 1971; Newell & Simon, 1972) concerned problems of a logical or mathematical nature that had in common a welldefined goal, as well as a well-known, transparent, static and thus more or less simple context. Constraints of the situation, possible ways of acting, and the effects of actions are immediately available. Such problem situations are valuable for research but do not sufficiently match the real world. Accordingly, he proposed a complex problemsolving environment presented as a computer simulation. This research made use of a computer environment in which the data of a small town were integrated (‘‘Lohhausen’’; cf. Do¨rner, Kreuzig, Reither, & Sta¨udel, 1983). For a simulated 10-year period, the problem solver served as the mayor and had to ensure that the small town would prosper by making decisions that influenced the town’s infrastructure, for example with respect to housing, schools, workplaces, local taxes, or medical care. Deciding in this context meant working in a system in which most

Problem solving in the mathematics classroom: the German perspective

actions had consequences that affected more than one element of the system. The approach focused on a new type of problems, which were not overtly mathematical in nature. However, these problems, too, required the application of knowledge that could be assigned as mathematical knowledge in many aspects. For a prospering town, for example, economical development was essential, which encompassed the town’s income and assets or the industrial production. Influencing this system positively meant to think in causes and effects, to identify adequate procedures, and to interpret intermediate results. The problem solver had to take into consideration that effects do not unfold immediately but with a time lag and that processes are often non-linear. The subjects took part in ten sessions in which they could decide on specific actions, which were put into the computer system. At the beginning of each section, they were informed about the consequences their changes caused in the town of Lohhausen. There are a number of issues, which were addressed in this study and turned out to be important for problemsolving processes. Most prominently, Do¨rner et al. (1983) mentioned self-regulation, which played a dominant role for successful problem solving. Comparing high-performing and low-performing subjects, the data suggested that successful problem solvers were more able to take advantage of information offered by the system, to include memorized facts, or to actively look for information. There was no evidence that motivation and general cognitive abilities as measured with regular tests correlated with successful problem solving in this dynamic system. This study did not only aim at learning effects from feedback but also at learning effects caused by training. However, the results suggested that there were no effects of a specific training. The participants subjectively felt that the training was helpful but objective measures did not support this impression (Do¨rner et al., 1983). The results showed that complex problem solving could be regarded as a difficult task. In particular, the application of multiple steps in a multidisciplinary context (‘‘crosscurricular competencies’’) had to be seen as a challenge for problem solvers. Moreover, the study provided evidence that dynamic problem solving such as thinking, acting, and arguing in a systemic context was difficult to learn. In this respect, problem solving in a realistic context turned out to be at least as sophisticated as mathematical problem solving.

3 Integrating problem solving in the German mathematics classroom: some remarks from a historical perspective Problem solving in Germany has roots in mathematics and psychology but it also found its way to schools and

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classrooms. In particular, it has never been questioned that problem solving as an integral part of doing mathematics should play a significant role in the mathematics classroom. However, in recent decades, problem-solving activities have undergone important changes with respect to the meaning it was assigned to by the curriculum and by mathematics teachers. This meaning became manifest in demanding tasks presented in the classroom (or the lacking of these tasks) but also in the way teachers actually worked with problems. Until the early 1960s teaching mathematics in German high schools was in the tradition of university mathematics. The famous mathematician Felix Klein (1849–1925) can be regarded as an indisputable representative for this tradition. His books on elementary mathematics (‘‘Elementarmathematik vom ho¨heren Standpunkt’’, e.g., Klein, 1924; English translation: elementary mathematics from an advanced standpoint: arithmetic, algebra, analysis resp. geometry) had a significant influence on the way mathematics instruction was seen in Germany. The mathematics classroom had to fulfill requirements that were suggested by mathematics as a scientific discipline. Correspondingly, textbooks from the 1950s and 1960s, particularly those for grades 11 through 13, contained numerous ambitious tasks. These tasks demanded for example competencies such as coherent argumentation or proof for their solution and were thus far more challenging than ordinary arithmetic or word problems. The tasks mentioned in the preceding paragraph had the potential for initiating problem solving but were rarely used in this respect. As mentioned above, problem solving in the mathematics classroom is not only a matter of demanding problems but also of the way teachers present these problems. It is probably this latter requirement which teachers did not meet at that time. Problem solving as an activity was more or less assigned to psychology but not the mathematics classroom. It may serve as an example that the title of Po´lya’s book ‘‘How to solve it’’ was translated into ‘‘Schule des Denkens’’ which means ‘‘School of Thought’’. The second author wishes to note that he bought this book in 1962 during his school days. The book had been a recommendation of his mathematics teacher, who was a person committed to the classroom. However, in retrospect, even this committed teacher was not able to let his students experience mathematics as problem solving. Although recognizing mathematics as a science which is based on problem solving was common among mathematics teachers, supporting students in their problem solving processes was less common in the 1950s and 1960s. It fits into this pattern that other books by Po´lya were translated into German (e.g., Po´lya, 1962, 1964, 1975), but received even less attention from teachers, mathematics educators, and mathematicians.

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However, the notion of problem solving became somewhat more prominent in the 1970s which can be seen in the mathematics curricula of that time. It should be noticed that here is a common tradition of the German (high school) mathematics classroom, however, Germany is a federal republic and the actual classroom work is highly determined by curricula specified by the state. Moreover, there were two parts of Germany until 1989 (West Germany and East Germany), which were under different authorities and had curricula which differed significantly. It is well known that these official and therefore intended curricula may not fully reflect the actual classroom instruction. Nonetheless they have an effect which makes it useful to have a look at the context in which problem solving was described in the curricula of the various states. The following citation may serve as an example for the way problem solving was seen in the Bavarian curriculum for elementary schools (grades 1 through 4): ‘‘The position of word problems in the 1974 Bavarian curriculum for elementary schools and their relationships to arithmetic are thoroughly reflected for each of the four grades. The relation between object, language, and number is demonstrated. Operators are a means useful for the methodological aspects of school instruction. Problem solving is finally outlined with useful and practical examples (Sattler, 1975; translation GT).’’ Thus, problem solving is explicitly mentioned in the curriculum. However, it is primarily determined by the application of mathematical knowledge rather than by the quality of the problem and its challenge. Nearly at the same time, namely in 1975, a text passage from a dissertation supports the importance of problem solving in the East German curriculum. Ro¨hr (1975) from the University of Leipzig (East Germany) comments on a textbook in the following way: ‘‘The author describes a possible way of attaining the educational objectives envisaged in the East German program for basic mathematics teaching by making the process of learning purposeful and creative. To accomplish this objective, basic heuristic statements on problem solving are used in accordance with Po´lya. The experiment showed that these methods may result in an improved performance in problem solving but need not do so in either case (translation GT).’’ These examples show that problem solving had a close connection to mathematics instruction at an early point in time (at least with respect to the international discussion). Moreover, the topic obviously became part of the curricula and gained increasing attention in both parts of Germany in the 1970s. It should be mentioned that the 1970s can be regarded as the starting point for a national competition for high-achieving students at the secondary level (‘‘Bundeswettbewerb Mathematik’’). The problems given in the course of this competition were (and still are) challenging problems in a tradition that takes into account high math-

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ematical standards as well as Po´lya’s definition of the notion. Research on problem solving in mathematics education was rarely undertaken in the 1970s and remained scarce in the following years. In particular, most research from mathematics educators in this area focused on psychological aspects and had its theoretical roots in psychology (e.g., concerning the Tower of Hanoi problem, Haussmann & Reiss, 1990; Reiss & Reiss, 1992). This research aimed at the process of developing scientific knowledge in the field of mathematics and sought to identify important factors of cognition in the process of problem solving. It should be added that despite the fact that problem solving was not explicitly addressed, German mathematics education broadly discussed the role of demanding problems in the mathematics classroom and gave recommendations how to integrate such problems in the mathematics classroom. In particular, elementary school education passed through a process which was influenced by Freudenthal (1973) and which emphasized realistic and challenging problems in the classroom that were supposed to foster the students’ active learning (e.g., Radatz & Schipper, 1983; Wittmann & Mu¨ller, 1990).

4 Problem solving in Germany: what are the interfering aspects? As the preceding sections of this article have suggested, problem solving has not been a central issue for mathematics education in Germany. There are a number of possible reasons for this which will be shortly discussed in the following. There is hardly any research concerning this point and so the thoughts of this section are highly speculative. We concentrate on a few aspects. Research: language barriers and comprehension problems. Certainly, it has been noticed in Germany that problem solving in the mathematics classroom was intensively discussed in the United States during the 1980s. However, the different meanings of the word problem solving and its direct German translation caused confusion and probably hindered a broader reception of the debate. The situation has changed recently. An important influence goes back to the Principles and Standards defined by the National Council of Teachers of Mathematics (NCTM) which explicitly indicate the nature of problem solving. Similar standards for school mathematics have been implemented in Germany and similar notions are used (cf. Sect. 5). The marginal role of problem solving in pre-service teacher education. Pre-service teacher education in

Problem solving in the mathematics classroom: the German perspective

mathematics is still dominated by teaching rather than by learning. Teachers are supposed to support their prospective students’ learning processes, however they rarely experience this support during their education. Listening to lectures is still the most important type of knowledge acquisition at the university level. It is difficult to change this situation in the near future. German universities have lectures in which hundreds of students participate as the dominant type for teaching in most subjects. Learning in small groups should nonetheless remain a most important objective of reforms. Problem solving as an activity for high-achieving students. Problem solving means high-quality teaching and learning of mathematics. Nonetheless, problems have an individual component that allows less able or interested students to experience other tasks as problems than their higher-achieving counterparts. A sound classroom implementation should take this into account and provide varying learning opportunities. But in this respect, things are changing. Mathematics instruction regarding the individual is becoming more prominent. Problems that are suited to the needs of a specific student are not too difficult to develop. It may serve as an example, that the mathematics competitions which formerly addressed only mathematically gifted students have been supplemented by competitions that address (nearly) all students in a classroom. The list of constraints for problem solving described here is far from being complete. Further research will need to address this topic and to seriously answer the question of possible obstacles in the classroom implementation.

5 Problem solving in Germany in a contemporary context The situation has changed in the past few years. There are two important reasons that fostered a broader discussion of problem solving in the classroom. Both aspects cannot be separated, as they are a consequence of the rather poor results German students showed when tested within the Program for International Student Assessment (PISA). In particular, their mathematical knowledge was of a largely algorithmic nature and hardly enabled them to apply it in specific contexts (Klieme, Neubrand, & Lu¨dtke, 2001). On the one hand, research was encouraged by these results to better investigate problem solving inside mathematics and outside mathematics. On the other hand, the school administration implemented standards for school mathematics. Both aspects will be discussed in the following paragraphs.

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5.1 Problem solving and the PISA study The examples from psychology presented in the first sections can hardly give an impression of the amount of research performed on problem solving in Germany and in the international context. Problem solving as information processing (Newell & Simon, 1972; Do¨rner, 1976) and the identification of differences between experts and novices (Chi, Glaser, & Farr, 1988; Plo¨tzner & Spada, 1994) were important strands. Moreover, problem solving and learning (Anderson, 1983) as well as developmental aspects (Oerter & Dreher, 1998) always played an important role. As a consequence it is not surprising that problem-solving skills have been regarded as important also in a school context. This importance was affirmed by including problem solving in the tests of PISA. The international study PISA provided a significant contribution to research on problem solving. PISA was initiated by the Organization for Economic Cooperation and Development (OECD) in order to compare competencies of 15-year-old students with respects to their literacy in reading, mathematics, and science. A German addendum encompassed problem solving in the first investigation dating back to 2000. Moreover, problem solving became an international component in PISA 2003. Problem solving as seen in the PISA testing is regarded to be spanning different domains and accordingly asks for cross-curricular competencies. The PISA 2000 cycle encompassed computer-assisted and traditional paper and pencil tasks, which came from a school-related content and non-school-related content and aimed at different psychological paradigms (e.g., inductive reasoning, dynamic systems, exploration in well-structured systems). All test items required knowledge from different domains for a correct solution. The results suggested that problem-solving competency had to be seen as a multifaceted construct. It included general strategies but also reading, science, and mathematics competencies (Funke, 2003). The results of the 2003 PISA cycle provided an even more concise picture. In this cycle, problem-solving competencies were tested internationally. Moreover, problem solving was assigned to cross-curricular competencies and restricted to analytical problem solving in the international context. There was a supplement exclusively for German schools. It encompassed test items, which asked for dynamical problem-solving competencies (Leutner, Klieme, Meyer, & Wirth, 2004). The analytical problems used in the international test described a clearly defined starting situation either verbally or by diagrams. All information was explicitly provided within the text or accessible by deductive arguments. The problem-solvers’ task was to analyze the situation and to

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develop a solution. In particular, the problems demanded decision-making, system analysis and design, and troubleshooting. Publicly available sample items show that they needed some mathematical knowledge for their solution but could not fully be assigned to mathematics (see Fig. 4 for an example; http://www.oecd.org). The data revealed that with respect to all OECD countries involved in the study, problem solving and mathematics (as tested in the PISA 2003 cycle) correlated nearly perfectly (r = 0.89 for students from all participating nations; r = 0.90 for the German sample; r = 0.98 with respect to the participating OECD countries). It should be taken into account that this was a latent correlation and accordingly from its nature higher than an ordinary correlation. Moreover, there were countries that ranked differently with respect to the mathematics and problem-solving

Fig. 4 Sample item from the PISA 2003 problem-solving test

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competencies of their students (e.g., Germany had relatively better scores in problem solving than in mathematics, the United States had better scores in mathematics compared to problem solving). However, the value must be interpreted in a way that problem solving and mathematics (both with respect to the PISA testing) described more or less the same construct. The question remains what constitutes the strong relation between mathematics and problem solving. The data of the PISA study unfortunately are not apt for an answer to this question. A German addendum to the PISA 2003 cycle encompassed questions on dynamical problem solving. It differed from analytical problem solving in a way that the information needed for solving a specific problem was not fully provided and the problems could not be solved by using exclusively deductive reasoning. It was, however, neces-

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sary to actively change the problem-solving situation and take into account changes caused by this interference (Leutner, Klieme, Meyer, & Wirth, 2004). Thus, dynamic problem solving was in a tradition, for example, with the concept of problem solving as used in the study of Do¨rner et al. (1983). In this part of the test, too, the data showed a strong correlation between mathematics and problem solving (r = 0.69; latent correlation, German sample). This correlation suggests that not only analytic but also dynamic problem solving requires competencies that have a common kernel with mathematics. Accordingly, analytic as well as dynamic problem solving could matter with regard to mathematics.1 5.2 Problem solving and the standards for school mathematics The results of the PISA study received enormous public attention in Germany. The fact that German students ranked rather poorly, scoring only at the average of the different OECD countries participating in the study, was of concern to the broad public. Accordingly, politicians felt that they had to react immediately. Germany is a federal republic, and the German school system lies in the responsibility of every single of the federal states. As a consequence of the PISA results, these federal states jointly discussed the problem and determined federal standards for different school subjects and in particular for school mathematics (Kultusminsterkonferenz, 2003, 2004a, 2004b). These standards for school mathematics show distinct similarities to the standards defined by the National Council of Teachers of Mathematics (2000). There are content standards (numbers, measuring, space and forms, functional connections, data and probability) and process standards (engaging in mathematics, problem solving, modeling, use of representations, argumentation and communication, dealing with formal and technical elements of mathematics). Problem solving is an explicit standard, which is (like the other standards) illustrated by sample items. Figure 5 gives an example of how problem solving in the context of the standards could be interpreted. The task is based on the description of the growth of a wood supply. 1

It might be worthwhile to add that problem solving was also a part of the Trends in International Mathematics and Science Study (TIMSS). However, the term was used in this study in a different sense, addressing arithmetic and simple word problems as mathematical problem-solving tasks whereas the PISA items set priorities on application oriented problem solving. It is not surprising that there was only a minor correlation between TIMSS and PISA results. However, it was the PISA study that thus could reveal shortcomings in the curricula.

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It starts at 80,000 m3 and has a growth rate of 2.5%. Part b. calls for representing the growth in a spreadsheet. Parts a (calculate the size of the wood supply after 2 years) and c (give another solution for b) are explicitly assigned to problem solving by the authors of the standards. This sample task is typical and gives an idea that problem solving is often regarded in the context of everyday applications within the standards. However, problem solving is not necessarily linked to applications of mathematics. Within the Principles and Standards as defined by NCTM, problem solving is meant to have students build new mathematical knowledge through problem solving, let them solve problems that arise in mathematics and in other contexts, encourage them to apply and adapt a variety of appropriate strategies, and foster their monitoring and reflecting on the process of mathematical problem solving (National Council of Teachers of Mathematics, 2000). This is quite similar to the objectives stated by the German Kultusministerkonferenz (KMK) which represents all government departments of education (Kultusministerkonferenz, 2003, 2004a, 2004b). According to these German standards problem solving encompasses working on given and individually posed problems, using heuristic strategies, principles, and tools, checking results for plausibility, and generating ideas for problem solving. As a consequence, one may state that there is now a curricular basis for problem solving as part of school mathematics in Germany. The KMK standards introduce a concept of problem solving that is not restricted to specific parts of the mathematics classroom but encourage it as an overarching activity. However, this broad concept of problem solving should be systematically illustrated by examples. Students and their teachers should be aware of a broad range of problems adequate for classroom use. Problem solving should not be regarded as an isolated activity but as a habit of mind in the mathematics classroom.

6 What will be the future of problem solving in Germany? Problem solving in the mathematics classroom has roots that were initiated by German mathematicians as well as by German psychologists. This tradition was not fully considered by mathematics educators and teachers in particular until the 1980s. Since then, things have been (slowly) changing. Students have become more and more encouraged to engage in problem-solving activities in the mathematics classroom, first initiated within the scientific mathematics education community, meanwhile supported by politicians and teachers. A ‘‘new culture of problems’’ is emerging and influences the school curricula. In a way,

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K. Reiss, G. To¨rner

440 Fig. 5 Sample item from the German standards for school mathematics

the situation in Germany now parallels that of the United States some years ago. Stanic and Kilpatrick (1989, p. 1) get to the point when stating: ‘‘Problems have occupied a central place in the school mathematics curriculum since antiquity, but problem solving has not. Only recently have mathematics educators accepted the idea that the development of problem-solving ability deserves special attention.’’ It is probably a relevant coincidence that these changes emerged in both countries when standards for school mathematics were introduced. Standards and curricula differ in an important aspect: curricula describe the input and thus specify the learning contents whereas standards describe the outcome and address what students should be able to accomplish (Klieme et al., 2004). Mathematical problems can be regarded as an aspect of the input as the citation above suggests, however problemsolving competencies are the outcome teachers wish their students to acquire. There is some evidence that this culture of problem solving is not only considered as important but that it is actually implemented in schools. Different indicators suggest this change in the mathematics classroom. On the one hand, there is a shift in mathematics textbooks for all grades from rather algorithmically oriented tasks to more demanding problems. This process started at the elementary school level but meanwhile there are outstanding examples of challenging textbooks for the secondary school level, too (e.g., Scha¨tz & Eisentraut, 2003). Moreover, problem solving in the classroom is reflected in nationwide or statewide tests. When implementing the KMK standards, the departments of education agreed to regularly check their fulfillment. The mathematical com-

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petencies of German students are accordingly subject to repeated testing in different grades. As an example the state of Bavaria may serve. There nearly all students in grades 2 and 3 as well as in grades 6 and 8 take part in standardized tests. The tests at the primary level are based on a competency model which explicitly includes problem solving (Reiss, 2004). In the course of implementing standards for school mathematics, the subject has sharpened its profile. It is reasonable to assume that the notion of problem solving will gain new meaning in this process. Fostering students’ mathematical competencies should primarily mean to help them improve their problem-solving skills. This requires good and challenging problems to be in the center of the mathematics classroom.

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