ࡱ>  @Tbjbj >؝؝G!!!8!<:",n|#####JJJmmmmmmm$oRqmP`IJPPm##%m8W8W8WP##m8WPm8W8Whi#" }b!HSh7ktm0,nhrTr$iripJK08WLMJJJmm$!V^! THE USE OF MATHEMATICAL MODELING TASKS TO DEVELOP CREATIVITY CHAN CHUN MING ERIC Abstract: In an effort to determine how children respond to newer pedagogic approaches, model-eliciting tasks were used as a tool for making children's mathematical thinking visible. This paper describes examples of children's creative responses as they engage in one such task. The children's responses matched some criteria of mathematical creativity which suggest that engaging children in mathematical modeling activities not only enables them to mathematize, it also develops them as creative thinkers. Keywords: Mathematical modeling, Model-eliciting task, Creative thinking, Problem solving, Problem-based learning INTRODUCTION The 1980s was known to be the decade of problem solving, and much was written about it in the light of educational reforms. However, despite reform attempts, Lester and Kehle (2003) lamented that nothing much had changed. Problem solving has continued to be characterized by direct linkages between givens and goals and the process being linear towards finding the correct answer. Pedagogical approaches are dominantly didactical because the teacher perceives himself or herself as being solely responsible for the teaching with the belief that without the direct instruction, children would not learn effectively. Instructions tend to focus on prescribed steps and accuracy in obtaining the correct answers. In one of my mathematics lessons where pupils were newly exposed to engaging model-eliciting tasks via a problem-based approach, the pupils constantly asked me Is this the correct answer? and Is this the right way? (Chan, 2007). As I tried to embrace educational reform efforts where constructivist and socio-cultural theories are said to underpin learning, I saw how entrenched the behavioristic aspects of conditioning had been; teacher-centered approaches conditioned children to seek assurance towards a sense of security, and getting correct answers implied having solved the problem. The children were not confident, and were in a state of wondering if they were doing it the way the teacher wanted it. If children continue to have that frame of mind, then it would limit creative thinking and hamper mathematical creativity as well. In an effort to involve children in mathematical modeling and for creativity thinking to surface, I arranged for Primary 6 pupils to be engaged in a series of model-eliciting tasks facilitated by their mathematics teacher. I argue in this paper that the use of such tasks facilitates meaningful problem solving, where opportunities abound for interpreting authentic problem situations, developing representational fluency, and generating solutions associated with different, new or unforeseen interpretations. IMPEDIMENTS TO CREATIVITY Whenever the notion of creativity is brought up for discussion amongst teachers, oftentimes the buzz about helping children to be creative thinkers seems to center more on disciplines that are not mathematics related. Children are encouraged to be imaginative in writing essays, be artistic in works of art or music, be innovative in designing webpages, or be resourceful in their Science or Social Studies projects. To be creative in mathematical problem solving is one aspect that is found wanting in mathematics classrooms. The lack of mathematical creativity is manifested in the use of mathematics along narrow domains, relying on routine processes and algorithms, and thinking in predominantly convergent ways about solving problems that could have been perpetuated by how mathematics has been taught and assessed in schools (Haylock, 1987). This slowness to embrace newer pedagogic approaches most probably is related to systemic and classroom reasons. For systemic reasons, tensions exist between what should be done to enact educational reform goals and the beliefs, constraints, limitations faced in the current system that restrain the teaching and learning of mathematics towards another level. These tensions are not new nor are they confined to any particular culture. Although education reforms continue to evolve, the lack of will-power and stamina to sustain the change coupled with the expectations to produce good academic results could easily leave teachers exhausted and reverting to conventional pedagogies. Henningsen and Stein (1997) attributed decline in mathematical activity to inappropriateness of task that have no mathematical substance, as well as classroom management problems and inappropriate amount of time as classroom reasons that inhibit mathematical thinking and reasoning which in a sense could also inhibit mathematical creativity. MATHEMATICAL CREATIVITY Mathematical creativity is a rather complex phenomenon. According to Haylock (1997), there is no one conclusive definition to it. However, he suggests two approaches for identifying creative thinking in problem solving; the overcoming of fixation or the breaking of a mental set, and determining the criteria for a product to be indicative of creative thinking such as flexibility, originality and appropriateness. Cropley (1992), in an attempt to define it likens it to the capacity to get ideas that are original, inventive and novel, while Pehkonen (1997) citing Kiesswetter (1983) associates flexible thinking to creative thinking, an aspect deemed to be the most important for successful problem solving. Silver (1997) portrays creativity as three core features; fluency, flexibility and novelty. Silver (ibid) also makes a distinction that contemporary views of creativity are not exceptional behaviours that are produced by geniuses using exceptional thought processes where creative works are seen as occasional bursts of insights. Rather, there is a creative disposition or orientation towards problem solving evidenced by using knowledge flexibly, reflections, and generative thinking. MATHEMATICAL MODELLING AND CREATIVITY Children need to interact with environments that can not only trigger critical and creative thinking but is sustained through discourse. Research has indicated that experts and novices tend to organize their mathematics knowledge around situations than general problem-solving heuristics or conventional math topics (Hall, 1999). In this paper, I advocate that children engage in modelling-eliciting tasks (in a problem-based approach) because the modeling process can foster critical and creative thinking. In other words, taking a problem-based approach through employing a model-eliciting task as a stimulus, and with the expectations of solving the problem collaboratively in small groups alongside the presence of teacher-scaffolding at strategic moments, children can be empowered to raise their levels of thinking and problem-solving capabilities. In mathematical modeling, children have to engage a model-eliciting task. A model-eliciting task is a problem which depicts a real-world situation. When actively engaged, the children exercise behaviours of problem interpretation, description, hypotheses, generalizations, and justifications as they try to make sense of the problem situation and model a mathematical solution (Doerr & English, 2003). Children are said to be mathematizing when they go through the problem-solving process iteratively. Socially they are also learning through their interaction and these approaches are deemed more favourable for mathematical development and generative thinking than the traditional approach. Thus, a mathematical modeling activity supports meaningful problem solving and creative thinking when children apply mathematics through redefining the problem situation towards problem resolution. MODEL-ELICITING TASK: THE BIGGEST BOX PROBLEM In this paper, I shall highlight the creative responses of several groups of pupils involved a modelling task that required them to make the biggest box (without cover). The description of the task is in Figure 1. Each group of four or five pupils was provided with the task sheet (Figure 1) and some stationery, and they proceeded to solve the problem after a brief orientation by me. For this task, it was not expected of the pupils to find the exact answer unless they knew calculus. It was a novel task because they had not encountered such a problem before and the task sheet did not provide any printed numerical values to begin with, quite unlike the structured word problems they were so used to doing. The problem context was realistic in that they had to work in small collaborative groups to present their findings to the teacher and me who were supposed to be the judges.    The typical way to approach this problem would be to apply mathematical ideas of nets and establish a relationship between how much of the lengths of the squares to cut out from the corners of the vanguard sheet and relate to the volume of the folded box (Figure 2). This was not an easy task as there were groups that looked nearly lost at first as they did not know how to begin. Some groups tried to maneuver and fold the vanguard sheets (without cutting) just to see if they would look like boxes while some questioned if boxes should be square or rectangular. Even though they had learned nets in their syllabus, it took a while before they sensed how what they had learned could be applied here to make a box by cutting four square corners away from the vanguard sheet. In the four examples that follow, I used Balka's (1974) criteria (cited in Mann, 2005) of mathematical creativity (Table 1) to match the problem solving actions. Table 1. Balka's (1974) Mathematical Creativity Criteria Criteria1Ability to formulate mathematical hypotheses concerning cause and effect in mathematical situations2Ability to determine patterns in mathematical situations.3Ability to break from established mind sets to obtain solutions in a mathematical situation4Ability to consider and evaluate unusual mathematical ideas, to think through the possible consequences for a mathematical situation5Ability to sense what is missing from a given mathematical situation and to ask questions that will enable one to fill in the missing mathematical information6Ability to split general mathematical problems into specific sub-problems Example 1 One of the challenges the groups faced was how they could convince themselves that they had obtained the biggest volume. If the length of the square cut-out could be a figure with 2 decimal places or 3 decimal places and so on, when would it then reach a stage before they knew they had obtained the ideal length to work towards the biggest box? A group had potentially maximized the volume using integers as dimension variables. They then went on to narrow the range of the lengths of the square cut-out up to one decimal place, and found that the volume of the box had increased marginally but it was nevertheless an increase that pleased them. They decided to stop. When they were questioned by the teacher (T) about considering cut-out lengths with 2 or more decimals places, a member (S2) recognized the limitation that the ruler could not accommodate the physical measuring of it. S2 acknowledged that they had thought about going into 2 decimal places but for practical reasons, decided not to pursue. TSo what is the size you are cutting out?S28.5TOne question to think about. What if you cut 8.5 something? Does it make a difference? (extending their thinking to use two or more decimal places)S2But it is very hard to measure.TYes. But have you all considered?S2Yeah, because the rule line does not have any smaller(scale to look at) The decision to justify for the biggest box was based on the systematic comparisons made with other sets of dimensions in relation to their respective volumes. This aspect of mathematical creativity suggests the ability to consider and evaluate unusual mathematical ideas (unusual in the sense that hardly any groups had thought about the possibility of going into two or more decimal places) and their consequences. However, this group was not done as they became more adventurous later (see example 2). Example 2 The group was inclined to explore further with cut-out lengths comprising 2 decimal places. After they had determined the maximized volume of the box they decided to increase the volume further by not wasting the four square cut-outs. Their revisions had obtained an ideal square cut-out earlier with a length of 8.35 cm each. One of the pupils said, You cut one square into four then you paste on top (of the box) They decided to use the four square cut-outs and sub-divided them into strips and attach them along each top edge of the box. This resulted in an even bigger volume due to an increase in height (see Figure 3). This shows an ability to break from established mindsets to obtain solutions in a mathematical situation. It also shows representational and strategic fluency.  Example 3 Another group had conflicting points of views when discussing the possibility of using decimal fractions in modeling the situation. The excerpt below captures how two members, S1 and S5, of the group grappled with that notion. S1 We can cut 0.1 cm. We can cut 0.1, then we get the biggest volume.S5 0.1? You got to be joking! How to get the biggest volume? You can cut out this much? Were supposed to convince the judge. Were going to present a box with 0.1 in height? Is it like a box? You go and cut! No, it wont be 0.1 in height.S1 Yes it is. It will. S5 Ridiculous! S1 proposed cutting out squares of 0.1 cm by 0.1 cm from the four corners of the vanguard sheet. This would leave a big square base area to multiply with the height of 0.1 cm to get a big volume. Mathematically, it made sense and was indeed quite a strategic move, but S5 was more pragmatic. She could not see the possibility of how a square of 0.1 cm by 0.1 cm could be physically cut, thus challenging S1 to show or prove to her. Both S1 and S5 presented plausible arguments; one mathematically, the other practically. Here it could be said that S1 had conjectured how the modeling could work which was a creative way of thinking through determining a certain pattern via manipulating the variables of the base area and height. S5s creative behaviour was observed as sensing that something was amiss and sought S1 to prove and justify it. Example 4 This final example shows how a group obtained the biggest box without any wastage (cut-outs). Relying on their understanding of nets, they devised a creative way of maximizing the volume. They divided the vanguard sheet into 9 equal squares. They conceived that they could make a net of a box with 9 squares. They therefore proceeded to make one square the base area (shaded square in middle picture of Figure 4), and attach the other 8 squares as the faces of the box with 2 squares per face. Although this might not be the biggest volume that one could get, it shows an ability to break from established mindsets to obtain solutions in a mathematical situation.  CLOSING REMARKS The examples above suggest that solving problems via a modelling approach can surface and help develop the mathematical creativity in children. Incidentally, Balkan's (ibid) criteria for mathematical creativity bear somewhat similar behavioral characteristics as mathematizing actions during mathematical modelling. Thus engaging children in mathematical modelling holds promise as a pedagogic approach that not only is consistent with current educational goals, there is also reason to believe it can develop children's mathematical creativity during problem solving. REFERENCES Balka, D. S. (1974). The development of an instrument to measure creative ability in mathematics. Dissertation Abstract International 36(01), 98. (UMI no. AAT 7515965) Chan, C. M. E. (2007). Using open-ended mathematics problems: A classroom experience (Primary). In C. Shegar & R. B. A. Rahim (Eds.), Redesigning pedagogy: Voices of Practitioners (pp. 129-146). Singapore: Pearson Education South Asia. Cropley, A. J. (1992). More ways than one: Fostering creativity. Norwood, New Jersey: Ablex Publishing Corporation. Doerr, H. M., & English, L. D. (2003). A modelling perspective on students' mathematical reasoning about data. Journal for Research in Mathematics Education, 34, 110-136. Hall, R (1999). Following mathematical practices in desing-oriented work. In C. Hoyles, C. Morgan, & G. Woodhouse (Eds.), Studies in mathematics education series: No.10. Rethinking the mathematics curriculum. Philadelphia: Falmer Press. Haylock, D. (1997). Recognising mathematical creativity in schoolchildren. ZDM, 29(3), 68-74. Haylock, D. W. (1987). A framework for assessing mathematical creativity in school children. Educational Studies in Mathematics, 18, 59-74. Henningsen, M., & Stein, M. K. (1997). Mathematical tasks and student cognition: Classroom-based factors that support and inhibit high-level mathematical thinking and reasoning. Journal for Research in Mathematics Education, 28(5), 524-549. Kiesswetter, K. (1983). Modellierung von Probleml seprozessen. In: Mathematikkunterricht 29(3), 71-101. Lester, F. K., & Kehle, P. E. (2003). From problem solving to modeling: The evolution of thinking about research on complex mathematical activity. In R. Lesh & H. Doerr, (Eds.), Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning and teaching (pp. 501-518). Mahwah, NJ: Lawrence Erlbaum Associates. Mann, E. L. (2005). Mathematical creativity and school mathematics: Indicators of mathematical creativity in middle school students. Unpublished doctoral dissertation, University of Connecticut. Pehkonen, E. (1997). The state-of-the-art in mathematical creativity. ZDM, 29(3), 63-67. Silver, E. A. (1997). Fostering creativity through instruction rich in mathematical problem posing and problem solving. ZDM, 29(3), 75-80. ABOUT THE AUTHOR Chan Chun Ming Eric Mathematics and Mathematics Education National Institute of Education Nanyang Technological University, Singapore E-mail: eric.chan@nie.edu.sg  PAGE 554 PAGE 554 PAGE 554 PAGE 558 PAGE 554    Chan Chun Ming Eric The Use of Mathematical Modeling Tasks to Develop Creativity PAGE 216 DG 9: Promoting Creativity for All Students in Mathematics Education, Section 3 PAGE 215 ICME 11, Mexico, 2008 Proceedings of the Discussing Group 9 : Promoting Creativity for All Students in Mathematics Education The 11th International Congress on Mathematical Education Monterrey, Mexico, July 6-13, 2008 PAGE 207 ICME 11, Mexico, 2008 Your team is participating in a math project competition where you will need to present your findings in two days' time. In the competition, each team has been given only two square sheets made of vanguard. The team can decide if they want to use one vanguard sheet for trial. The team is supposed to make the biggest box (volume) using only ONE vanguard sheet. How would your team plan to solve the problem of making the biggest box? Show in detail how you reach a solution to convince the judge. (Note that your box must have a base but need not have a cover) Figure 1. The problem task 8.35 cm x 4 = 33.4 cm The 4 squares make up 33.4cm when joined 33.4cm Length of box is 33.3 cm Cut horizontally into 4 strips and patch on top of faces of the box 33.4cm Figure 3. Maximizing the volume of a box Figure 4. The making of a box without "wastage" Figure 2. 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