Problem-posing research in mathematics education: new questions and directions
Florence Mihaela Singer, Nerida Ellerton & Jinfa Cai
Educational Studies in Mathematics An International Journal ISSN 0013-1954 Educ Stud Math DOI 10.1007/s10649-013-9478-2
1 23
Your article is protected by copyright and all rights are held exclusively by Springer Science +Business Media Dordrecht. This e-offprint is for personal use only and shall not be selfarchived in electronic repositories. If you wish to self-archive your work, please use the accepted author’s version for posting to your own website or your institution’s repository. You may further deposit the accepted author’s version on a funder’s repository at a funder’s request, provided it is not made publicly available until 12 months after publication.
1 23
Author's personal copy Educ Stud Math DOI 10.1007/s10649-013-9478-2
Problem-posing research in mathematics education: new questions and directions Florence Mihaela Singer & Nerida Ellerton & Jinfa Cai
# Springer Science+Business Media Dordrecht 2013
Abstract As an introduction to the special issue on problem posing, the paper presents a brief overview of the research done on this topic in mathematics education. Starting from this overview, the authors acknowledge important issues that need to be taken into account in the developing field of problem posing and identify new directions of research, some of which are addressed by the collection of the papers included in this volume. Keywords Problem posing . Field of mathematics education research . Problem solving . Instruction In the 1980s, a well-known philosopher of science, Karl Popper, and a Nobel Prize laureate, John Eccles, worked together to explain the process of scientific discovery. They concluded “that the process of understanding and the process of the actual production or discovery of… [theories, scientific laws, etc.] are very much alike. Both are making and matching processes” (Popper & Eccles, 1977, p. 461). This vision has even older roots: the eminent mathematician Jacques Hadamard (1945) remarked: Between the work of the student who tries to solve a problem in geometry or algebra and a work of invention, one can say that there is only a difference of degree, a difference of level, both works being of a similar nature. (p. 104) Hadamard was emphasizing a vision in which problem solving is meaningful if it engenders invention or at least reveals new issues. And several years earlier, Einstein and Infeld (1938) maintained that posing an interesting problem is more important than solving it. F. M. Singer (*) University of Ploiesti, Ploiesti, Romania e-mail:
[email protected] N. Ellerton Illinois State University, Normal, USA J. Cai University of Delaware, Newark, USA
Author's personal copy F.M. Singer et al.
Looking further back into the history of human thought, we find that Socrates (469 BCE– 399 BCE) established an efficient method of learning through a continuous dialog based on posing and answering questions to stimulate critical thinking and illuminate ideas. The focus on the nature of critical thinking has continued ever since and is a contemporary issue that has become more and more important in education. This brings us to our point of interest, which is how teaching and learning might be organized in order that it better responds to the educational needs of our contemporary society. Leaving the world of the great thinkers and entering everyday life, we find the need for every person to be aware of the problems he/she encounters in order to be in a better position to solve them. As Kilpatrick (1987) noted: “In real life outside of school […] many problems, if not most, must be created or discovered by the solver, who gives the problem an initial formulation” (p. 124). More than ever, today's dynamic society requires school graduates who are able to adapt to new, frequently unpredictable situations (such as changing jobs, changing homes, and changing professions many times during a lifetime) and to make knowledgeable decisions in those situations. Knowing how to identify and formulate mathematical problems can contribute to making such good decisions. The ideas discussed above have a common focus on problem posing as an important vector for learning. During the last 25 years, the educational potential of integrating problem posing at various levels of formal instruction has been advocated by an increasing number of educators and researchers (see, for example, Cai, 2005; Ellerton, 1986; Silver & Cai, 1996). Problem posing improves students' problem-solving skills, attitudes, and confidence in mathematics, and contributes to a broader understanding of mathematical concepts and the development of mathematical thinking (Cai & Cifarelli, 2005; English, 1998; Silver, 1994, 1997; Singer & Moscovici, 2008). Problem posing can also serve as a means of enhancing creativity (e.g., Silver, Kilpatrick, & Schlesinger, 1990). Most often, following Torrance's (1974) influence, the study of creativity in mathematics has taken into consideration three categories, mostly with reference to problem solving: fluency, flexibility, and originality (e.g., Ervynck, 1991; Kontorovich, Koichu, Leikin, & Berman, 2012; Leikin & Lev, 2007; Leung, 1997; Silver, 1997). Less frequently, they have been used to characterize creativity in problem posing (e.g., Yuan & Sriraman, 2010). New approaches start from a model of developing creativity in organizational contexts and examine problem posing from the perspective of cognitive flexibility (e.g., Singer, Pelczer, & Voica, 2011; Singer, 2008; Voica & Singer, 2012). From a cognitive point of view, problem posing can sustain the natural endowment of the human mind for bootstrapping (Singer, 2009) and thus may contribute to developing a context for authentic stimulating activities in the classroom. According to Kilpatrick (1987), problem posing contributes to the development of mathematical aptitude and learning autonomy (e.g., Kilpatrick, 1987; Mamona-Downs & Downs, 2005; Silver & Cai, 1996). However, the topic of posing problems has largely remained outside the vision and interest of the mathematics education community. At its heart, problem posing is based on inquiry and thus affords contexts where students can discuss and evaluate a variety of approaches that have been developed and presented by classmates. Within a climate in which students are invited to give their contribution by adding to, enhancing, or changing the problem, the social component is likely to emerge (English, 2003). Although various studies have analyzed the reasoning processes of students as they solve well-structured problems, only a few studies (e.g., Cifarelli & Cai, 2005; Nohda, 1995; Ponte, 2007) have focused on illustrating and explaining characteristics of the solvers' mathematical explorations in open-ended problem situations. More studies are needed on the processes that students use when they are working in such situations or in
Author's personal copy Problem-posing research: new questions and directions
situations in which some aspects of the task are unspecified, and the solver is required to reformulate the problem statement in order to develop solutions (Singer, Ellerton, Cai, & Leung, 2011). Some studies have focused on structural questions associated with the issue of when problem-posing sessions should take place and what their format might be. According to Silver (1994), problem-posing activities might be organized before (pre-solution), during (within-solution), or after (post-solution) the problem-solving process. Problem-posing situations can be free, semistructured, or structured (Stoyanova & Ellerton, 1996). They may emphasize the formulation of a key problem that could trigger an extended and productive mathematical activity, or they may function as a constant process of posing subsidiary questions (Christou, Mousoulides, Pittalis, Pitta-Pantazi, & Sriraman, 2005; Ponte & Matos, 1992). Challenges remain, however, in defining the characteristics of problem posing, identifying possible relationships between the various subcategories of problem posing, and investigating possible interrelationships and interdependence between problem posing and problem solving in theory and in practice (Singer et al., 2011; Stoyanova & Ellerton, 1996). Various ways to generate new problems have been suggested by authors like Silver, Mamona-Downs, Leung, and Kenney (1996), who put forward the following strategies: constraint manipulation (i.e., systematic manipulation of the task conditions or implicit assumptions), goal manipulation (i.e., manipulation of the goal of a given or previously posed problem where the assumptions of the problem are accepted with no change), symmetry (i.e., a symmetric exchange between the existing problem's goal and conditions), and chaining (i.e., expanding an existing problem so that a solution to the new problem would require solving the existing one first). Experiments have shown that systematic training, focused on problem modification through the use of various representations, and addition or exclusion of some elements of the initial problem (such as operations, data, constraints, questions, etc.), accompanied by comparisons between the resulting problems in order to assess similarities and differences among them, as well as the analysis of incomplete or redundant problems, can raise students' awareness of the consistency and meaningfulness of the problems (Singer, 2007, 2010). Some of these aspects remain contentious and require further investigation. Additional research is still needed to provide evidence supporting these and other advantages of problem-posing approaches in the classroom. Well-designed problem-posing tasks for students also require skilled teachers who can manage the complexities of such contexts. Moreover, in-service and prospective mathematics teachers should be offered opportunities to pose their own problems. The successful use of authentic mathematical activities in teaching requires teachers who have experience with such activities themselves as learners of mathematics (e.g., Brown & Walter, 2005; Crespo & Sinclair, 2008; Ellerton & Clarkson, 1996). Indeed, many studies have reported that, frequently, both students and teachers pose problems that are ill formulated or lack cognitive demand (e.g., Harel, Koichu, & Manaster, 2006; Silver & Cai, 1996, 2005). Investigations of problems posed by students and teachers in classrooms have provided insight into the relationships between mathematical knowledge, skills, and processes (Ma, 1999; Verschaffel & De Corte, 1997). Problem-posing sessions in the classroom, for example, allow the development of tasks that integrate a variety of activities that specifically address students' motor, visual, and verbal skills, as well as various types of transfer between these domains, to allow a variety of representations and representational changes (Singer, 2007, 2012) in students. There is therefore a need to study problem-posing techniques that are already practiced in some classrooms, in order to analyze and extend those strategies that proved to be effective.
Author's personal copy F.M. Singer et al.
Scholars have recognized that problem posing is an important part of mathematical activity, yet research on problem posing has not been a major focus in mainstream mathematics education research. Thus, the field would benefit from systematizing the theoretical curriculum and pedagogical foundations for problem posing, as well as the empirical findings accumulated to date in problem-posing research. Our explorations into this burgeoning field of mathematics education research show that the time has come for more systematic analyses that can organize the research and theory of problem posing as well as its applications to the practice of teaching. At present, the field of problem posing is still very diverse and lacks definition and structure. We are committed to taking on the challenge of both witnessing and contributing to foundational research on problem posing which will help address this lack of definition and structure. However, we consider that problem-posing research is an emerging force within mathematics education; imposing a formal structure on this societally based developing field would not be in accordance either with the field itself or with the theories of paradigm change in scientific communities (Kuhn, 1962). Consequently, we want to promote a large spectrum of opinions and visions that will naturally contribute to both the development and the structure of the field. The idea for this special issue was born in 2009 at a PME 33 multinational working group titled: Problem Posing in Mathematics Learning: Establishing a Theoretical Base for Research. The working session, coordinated by Jinfa Cai, Nerida Ellerton, Mitsunori Imaoka, Ildiko Pelczer, Florence Mihaela Singer, and Cristian Voica, engaged participants in discussions on the following topics:
& & & &
Problem posing as an integral component of school mathematics Contrasting the cognitive components of problem posing and problem solving in mathematical thinking Problem posing and discourse in mathematics classrooms, and Problem-posing processes and how these relate to creativity
These discussions identified some solutions for approaching problem posing, but raised many new questions. Consequently, the debate continued with a research forum at PME 35 in 2011, titled: Problem Posing in Mathematics Learning and Teaching: a Research Agenda, coordinated by Florence Mihaela Singer, Nerida Ellerton, Jinfa Cai, Craig Cullen, Ildiko Pelczer, Cristian Voica, and Eddie Leung. The research forum stimulated further discussion and research about ways in which problem posing could become a more natural and integral part of mathematics classrooms at all levels. The two 90-min sessions of the research forum engaged considerable interest among researchers who actively contributed to presentations, discussions, and conclusions. The forum addressed questions like:
& & &
How do problem posing and problem solving interact in the course of ongoing mathematical exploration? How could students' learning be improved through engaging them in mathematical exploration and problem posing in open-ended problem situations? Is it feasible to use problem posing as an assessment tool?
Practical issues for using mathematical exploration and problem posing as pedagogical strategies for improving students' learning in the classroom were also examined. The connections between exploring open-ended problems and learning, as well as the conceptual
Author's personal copy Problem-posing research: new questions and directions
benefits, limits, and difficulties of having students engaged in mathematical explorations, were discussed extensively. During the forum, the audience was challenged to take part in the inquiry involving aspects concerning teachers' roles and students' roles in problemposing sessions, on the one hand, and the nature of the tasks, on the other hand. This special issue proposes some answers to the above questions. We have structured the volume around the following main topics, which emerged from both empirical and theoretical research: Research-based Insights for Designing Problem-Posing Activities, ResearchBased Commentaries on the Nature of Problem Posing, and Problem Posing as a Research and Instruction Tool. The diversity of topics is complemented by the diversity of research methods and samples. Techniques, challenges, and strategies of implementing a mathematical problem-posing approach in the classroom are reported for samples of students from all levels of schooling, as well as of college students, and preservice and in-service teachers. New directions for research and practice emerge from the collection of the papers included in this volume. One of the aims of the volume is to provide a research base for practitioners who themselves pose and/or who encourage their students to pose mathematical problems. A first direction refers to designing problem-posing tasks by starting from problem-solving frameworks, or by meaningful situations involving the use of cultural artifacts, or by reinforcing a certain curriculum. Papers by Singer and Voica, by Olson and Knott, and by Bonotto are included in this section. A second direction of research is focused on understanding the nature of problem posing through comparative studies, with papers by Cai, Moyer, Wang, Hwang, Nie, and Garber, and by Koichu and Kontorovich. The study of creative approaches to mathematics brings another direction of research, which considers the relationships between students' mathematical content knowledge and their problem-posing abilities in mathematics. Papers by Ellerton, by Leung, and by Van Harpen and Presmeg are presented in this section. The effect of problem-posing-based learning situations on the instructional process constitutes a fourth direction of research reflected in this collection and includes papers by Hošpesová and Tichá, and by da Ponte and Henriques. Last but not least, Edward Silver provides a scholarly critique and overview of the papers in this special issue and reflects on his views of the future of the field. As we stressed at the beginning of this introduction, problem posing is an old issue. What is new is the awareness that problem posing needs to pervade the education systems around the world, both as a means of instruction (meant to engage students in genuine learning activities that produce deep understanding of mathematics concepts and procedures) and as an object of instruction (focused on developing students' proficiency in identifying and formulating problems from unstructured situations) with important targets in real-life situations. This special issue aims to pave the way in the directions we have discussed above by sharing experiences and expertise to develop better thinkers and better schools for the twenty-first century.
References Brown, S., & Walter, M. (1983/2005). The art of problem posing (3rd ed.). Mahwah, NJ: Lawrence Erlbaum Associates. Cai, J. (2005). U.S. and Chinese teachers' knowing, evaluating, and constructing representations in mathematics instruction. Mathematical Thinking and Learning: An International Journal, 7(2), 135–169. Cai, J., & Cifarelli, V. (2005). Exploring mathematical exploration: how do two college students formulate and solve their own mathematical problems? Focus on Learning Problems in Mathematics, 27(3), 43–72. Christou, C., Mousoulides, N., Pittalis, M., Pitta-Pantazi, D., & Sriraman, B. (2005). An empirical taxonomy of problem posing processes. Zentralblatt für Didaktik der Mathematik, 37(3), 149–158.
Author's personal copy F.M. Singer et al. Cifarelli, V., & Cai, J. (2005). The evolution of mathematical explorations in open ended problem solving situations. The Journal of Mathematical Behavior, 24, 302–324. Crespo, S., & Sinclair, N. (2008). What makes a problem mathematically interesting? Inviting prospective teachers to pose better problems. Journal of Mathematics Teacher Education, 11(5), 395–415. Einstein, A., & Infeld, L. (1938). The evolution of physics. New York: Simon & Schuster. Ellerton, N. F. (1986). Children's made-up mathematics problems—a new perspective on talented mathematicians. Educational Studies in Mathematics, 17(3), 261–271. Ellerton, N. F., & Clarkson, P. C. (1996). Language factors in mathematics teaching. In A. J. Bishop, K. Clements, C. Keitel, J. Kilpatrick, & C. Laborde (Eds.), International handbook of mathematics education (pp. 987–1053). Netherlands: Kluwer. English, L. D. (1998). Children's problem posing within formal and informal contexts. Journal for Research in Mathematics Education, 29(1), 83–106. English, L. D. (2003). Problem posing in the elementary curriculum. In F. K. Lester Jr. & R. I. Charles (Eds.), Teaching mathematics through problem solving: Prekindergarten—grade 6 (pp. 187–198). Reston: NCTM. Ervynck, G. (1991). Mathematical creativity. In D. Tall (Ed.), Advanced mathematical thinking (pp. 42–53). Dordrecht: Kluwer. Hadamard, J. W. (1945). Essay on the psychology of invention in the mathematical field. Princeton: Princeton University Press (page ref. are to Dover edition, New York 1954). Harel, G., Koichu, B., & Manaster, A. (2006). Algebra teachers' ways of thinking characterizing the mental act of problem posing. In I. J. Novotna, H. Moraova, M. Kratka, & N. Stehlikova (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 241–248). Prague: Charles University. Kilpatrick, J. (1987). Problem formulating: Where do good problems come from? In A. H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 123–147). Hillsdale: Lawrence Erlbaum Associates. Kontorovich, I., Koichu, B., Leikin, R., & Berman, A. (2012). An exploratory framework for handling the complexity of mathematical problem posing in small groups. The Journal of Mathematical Behavior, 31(1), 149–161. Kuhn, T. S. (1962/1969). The structure of scientific revolutions. International Encyclopedia of Unified Science. Volume 2, no. 2. Retrieved from http://turkpsikiyatri.org/arsiv/kuhn-ssr-2nded.pdf Leikin, R., & Lev, M. (2007). Multiple solution tasks as a magnifying glass for observation of mathematical creativity. In Proceedings of the International Group for the Psychology of Mathematics Education, Seoul, Korea, 8–13 July 2007 (pp. 161–168). Seoul, Korea: The Korea Society of Educational Studies in Mathematics. Leung, S. S. (1997). On the role of creative thinking in problem posing. Zentralblatt für Didaktik der Mathematik, 97(3), 81–85. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States. Mahwah: Lawrence Erlbaum Associates. Mamona-Downs, J., & Downs, M. (2005). The identity of problem solving. The Journal of Mathematical Behavior, 24, 385–401. Nohda, N. (1995). Teaching and evaluating using “open-ended problems” in the classroom. Zentralblatt für Didaktik der Mathematik, 27(2), 57–61. Ponte, J. (2007). Investigations and explorations in the mathematics classroom. ZDM, 39, 419–430. Ponte, J., & Matos, J. F. (1992). Cognitive processes and social interactions in investigation activities. In J. Ponte et al. (Eds.), Mathematical problem solving and new information technologies: Research in contexts of practice (pp. 239–254). Berlin: Springer. Popper, K. R., & Eccles, J. A. (1977). The self and its brain: An argument for interactionism. Berlin: Springer. Silver, E. A. (1994). On mathematical problem posing. For the Learning of Mathematics, 14(1), 19–28. Silver, E. A. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing. ZDM, 29(3), 75–80. Silver, E. A., & Cai, J. (1996). An analysis of arithmetic problem posing by middle school students. Journal for Research in Mathematics Education, 27(5), 521–539. Silver, E. A., & Cai, J. (2005). Assessing students' mathematical problem posing. Teaching Children Mathematics, 12(3), 129–135. Silver, E. A., Kilpatrick, J., & Schlesinger, B. (1990). Thinking through mathematics: Fostering inquiry and communication in mathematics classrooms. New York: The College Board. Silver, E. A., Mamona-Downs, J., Leung, S., & Kenney, P. A. (1996). Posing mathematical problems: an exploratory study. Journal for Research in Mathematics Education, 27(3), 293–309. Singer, F. M. (2007). Beyond conceptual change: Using representations to integrate domain-specific structural models in learning mathematics. Mind, Brain, and Education, 1(2), 84–97.
Author's personal copy Problem-posing research: new questions and directions Singer, F. M. (2008). Enhancing transfer as a way to develop creativity within the dynamic structural learning. In R. Leikin (Ed.), Proceedings of the 5th International Conference on Creativity in Mathematics and the Education of Gifted Students (pp. 223–230). Tel Aviv: CET. Singer, F. M. (2009). The dynamic infrastructure of mind—a hypothesis and some of its applications. New Ideas in Psychology, 27, 48–74. Singer, F. M. (2010). Children's cognitive constructions: From random trials to structures. In J. A. Jaworski (Ed.), Advances in sociology research (Vol. 6, pp. 1–35). Hauppauge: Nova. Singer, F. M. (2012). Boosting the young learners' creativity: Representational change as a tool to promote individual talents (plenary lecture). In The 7th International Group for Mathematical Creativity and Giftedness (MCG) International Conference Proceedings (pp. 3–26). Busan, South Korea: MCG. Singer, F. M., Ellerton, N., Cai, J., Leung, E. (2011). Problem posing in mathematics learning and teaching: A research agenda. In B. Ubuz (Ed.), Developing mathematical thinking. Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 137–166). Ankara, Turkey: PME. Singer, F. M., & Moscovici, H. (2008). Teaching and learning cycles in a constructivist approach to instruction. Teaching and Teacher Education, 24(6), 1613–1634. Singer, F. M., Pelczer, I., & Voica, C. (2011). Problem posing and modification as a criterion of mathematical creativity. In T. Rowland & E. Swoboda (Eds.) Proceedings of the 7th Conference of the European Society for Research in Mathematics Education (CERME 7) University of Rzeszów, Poland, 9–13 February, 2011. Stoyanova, E., & Ellerton, N. F. (1996). A framework for research into students' problem posing in school mathematics. In P. Clarkson (Ed.), Technology in mathematics education (pp. 518–525). Melbourne: Mathematics Education Research Group of Australasia. Torrance, E. P. (1974). Torrance tests of creative thinking: Norms and technical manual. Bensenville: Scholastic Testing Services. Verschaffel, L., & De Corte, E. (1997). Teaching realistic mathematical modeling in the elementary school. A teaching experiment with fifth graders. Journal for Research in Mathematics Education, 28, 577–601. Voica, C., & Singer, M. (2012). Creative contexts as ways to strengthen mathematics learning. In M. Aniţei, M. Chraif & C. Vasile (guest Eds.), Procedia—Social and Behavioral Sciences, PSIWORLD 2011 (vol. 33, 538–542). Available at http://www.sciencedirect.com/science/article/pii/S1877042812001875. Accessed 3 Oct 2013. Yuan, X., & Sriraman, B. (2010). An exploratory study of relationships between students' creativity and mathematical problem posing abilities. In B. Sriraman & K. Lee (Eds.), The elements of creativity and giftedness in mathematics (pp. 5–28). Rotterdam: Sense Publishers.
