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On Mathematical Problem Posing* On Mathematical Problem Posing* EDWARD A. SILVER In mathematics classes at all levels of schooling in all itself an exercise in exploring, conjecturing, examining, and countries of the world, students can be observed solving testing-all aspects of problem solving .. 1 asks should be problems .. The quality and authenticity of these mathemat­ created and presented that are accessible to students and ics problems has been the subject of many discussions and ~xtend their knowledge of mathematics and problem solv­ debates in recent years. Much of this attention has resulted ing. Students should be given opportunities to formulate in a rich, more diverse collection of problems being incor­ problems from given situations and create new problems by porated into school mathematics curricula. Although the modifying the conditions of a given problem [NCIM, problems themselves have received much scrutiny, less 1991, p. 95] attention has been paid to diversifying the sources for the Despite this interest, however, there is no coherent, com­ problems that students are asked to consider in school Stu­ prehensive account of problem posing as a prut of mathe­ dents are almost always asked to solve only the problems matics curriculum and instruction nor has there been sys­ that have been presented by a teacher or a textbook Stu­ tematic research on mathematical problem posing [Kil­ dents are rarely, if ever, given opportunities to pose in patrick, 1987] For the past several years, I have been some public way their own mathematics problems. Tradi­ working with colleagues and students on a number of tional transmission/reception models of mathematics investigations into various aspects of problem posing .. [1] instruction and learning, which emphasized students pas­ Our experiences in studying mathematical problem posing, sively receiving knowledge as a result of transmission and our reading of the work of others interested in this area teaching, were compatible with a pedagogy that placed the have formed the basis for this paper. responsibility for problem posing exclusively in the hands This paper begins with a brief introduction to the types of teachers and textbook authors On the other hand, con­ of activities and cognitive processes that have been temporary constructivist themies of teaching and learning referred to as problem posing, and then identifies and dis­ require that we acknowledge the importance of student­ cusses various perspectives from which one can view the generated pmblem posing as a component of instructional role and place of problem posing in the school mathemat­ activity. ics curriculum. Kilpatrick has argued that "problem formu­ Problem posing has been identified by some distin­ lating should be viewed not only as a goal of instruction guished leaders in mathematics and mathematics education but also as means of instruction" [1987, p .. 123], and both as an important aspect of mathematics education [e.g, views of problem posing will be evident in this paper The Freudenthal, 1973; Polya, 1954].. And problem posing has nature and findings of some research related to mathemati­ recently begun to receive increased attention in the litera­ cal problem posing is also discussed in order to character­ ture on curricular and pedagogical innovation in mathe­ ize some of the available research evidence associated with matics education. In the United States, fOr example, recent each of the perspectives discussed and to suggest some reports, such as the Curriculum and evaluation standards important issues in need of further investigation. To illus­ for school mathematics [NCTM, 1989] and the Profe,ion­ trate the international interest in mathematical problem al standards for teaching mathematics [NCTM, 1991], posing, examples of research and opinion from around the have called for an increase in the use of problem-posing world are discussed activities in the mathematics classroom. Both reports have suggested the inclusion of activities emphasizing student­ What is mathematical problem posing? generated problems in addition to having students solve Problem posing refers to both the generation of new prob­ pre-formulated problems, as is clearly illustrated in the fol­ lems and the re-formulation, of given problems. Thus, pos­ lowing excerpt from the Profes'sional standards for teach­ ing can occur before, during, or after the solution of a ing mathematics: problem I eaching mathematics from a problem-solving perspective One kind of problem posing, usually referred to as prob­ entails more than solving nonroutine but often isolated prob­ lem formulation or re-formulation, occurs within the pro­ lems or typical textbook types of problems It involves the cess of problem solving When solving a nontrivial prob­ notion that the very essence of studying mathematics is lem a solver engages in this form of problem posing by recreating a given problem in some ways to make it more accessible for solution.. Problem formulation represents a *This paper was presented as a plenary address at the seventeenth annual kind of problem posing process because the solver trans­ meeting of the International Group for the Psychology of Mathematics fonns a given statement of a problem into a new version Education, Tsukuba, Japan, July 1993 Preparation of the paper was supported by National Science Foundation grant MDR-8850580 that becomes the focus of solving, Problem formulation is Opinions expressed are those of the author and do not necessarily reflect related to planning, since it may involve posing problems those of the Foundation For the Learning of Mathematics 14, 1 (February, 1994) FL M Publishing Association, Vancouver. British Columbia, Canada 19 that represent subgoals for the larger problem. Polya's tives from which to view the importance and role of math­ heuristic advice "Think of a related, more accesstble prob­ ematical problem posing as an object of pedagogical and lem," suggests ;nother way in which problem formulation research attention. The purpose here is not to focus on involves problem posing. If the source of the original prob­ sharp distinctions among these perspectives, since they_ are lem is outside the solver, the problem posing occurs as the not mutually exclusive, but rather to use these perspecttves given problem is reformulated and "personalized" through as lenses through which to view various research studies the process of re-formulation. The operative question that and instructional interventions that have been undertaken stimulates this form of posing is: how can I formulate th1s Problem posing as a feature of creative activity or excep­ problem so that it can be solved? tional mathematical ability At least since Duncker's [1945] observation that prob­ Problem posing has long been viewed as a characteristic of lem solving consists of successive re-formulations of an creative activity or exceptional talent. For example, initial problem, problem formulation has been extensively Hadamard [1945] identified the ability to find key research studied by psychologists interested in understandmg. com­ questions as an indicator of exceptional mathematical tal­ plex problem solving. According to contemporary ~nfor­ ent Related observations have been made about profes­ mation-processing models of complex problem solvmg, a sionals in various science fields [e.g., Mansfield & Busse, problem is solved by establishing ~ series _of successtvely 1981].. Similarly, Getzels and Csikszentmihalyi [1976] more refined problem representation whtch mcorporate studied artistic creativity and characterized problem find­ relationships between the given information and the ing as being central to the creativ~ artistic expe!i~nc~ desired goal, and into which new infor~ati~n i~ added as The apparent link between posmg and creatlV!ty !S cleat subgoals are satisfied One of the maJor fmdmgs of an from the fact that posing tasks have been included in tests extensive body of research on the differences between designed to identify creative individuals .. For example, experts and novices in a variety of complex task domams Getzels and Jackson [1962] developed a battery of tests to is that experts tend to spend considerable time engaging in measure creativity, of which one task asked subjects to problem formulation and re-formulation, usually engagmg pose mathematical problems that could be answered using in qualitative rather than quantitative analysis, in contr:ast information provided in a set of stories about real-world to novices who spend relatively little time in formulat10n situations. Getzels and Jackson scored the subjects' prob­ and re-formulation [Silver & Marshall, 1989] For relative­ lems according to the complexity of the procedures that ly simple problems, problem formulation rna~ occur pri­ would be used to obtain a solution (i.e , the number and marily in the early stages of problem solvmg, but 1n type of arithmetic operations used), and they used the extended mathematical investigations, "problem formula­ results as a measure of creativity. Balks [1974] also asked tion and problem solution go hand in hand, each eliciting subjects to pose mathematical problems that could be the other as the investigation progresses" [Davis, 1985, p answered on the basis of information provided in a set of 23] stories about real-world situations. Analysis of the Not all problem posing occurs within the process of responses attended to three aspects: fluency, flexibility, solving a complex problem Problem posing can also occur and originality. Fluency refers
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