How to Share an Island: Divergent Thinking and Creativity

Nikita Patterson and Darryl Corey

Introduction and Motivation.

Divergent thinking, namely producing as many ideas as possible about a problem, is often not a regular part of mathematics instruction in the United States. Often times, it is the lack of a teacher’s confidence in his/her own ability to think ‘outside the box’ for additional solution methods that prevents the introduction of such activities to students” (Corey & Unal, 2004, pg 1). Moreover, most problems in mathematics yield only one correct answer, thus they are highly likely to discourage students from exploring or discussing diverse ideas while solving problems (Kwon, Park, & Park, 2006). However, both state and national standards for mathematics require that we “apply and adapt a variety of appropriate strategies” when engaging in mathematical problem solving (NCTM, 2009). We can improve our own learning by becoming aware of our own thinking as we read, write, and reflect on our own problem solving process while seeking different solution strategies to the same problem. As teachers, we can promote this awareness directly by informing students about a variety of strategies for solving mathematics problems. This can help to construct classroom environments that focus on strategic learning that is both flexible and creative. In most instances, teachers are the ones who shape students’ perception of and performance in mathematics. Improving mathematics performance and understanding are very important goals for the education community. One way to accomplish this goal is to improve students’ ability to “think outside the box” by engaging them in divergent thinking while solving problems in mathematics (Corey, Unal, & Jakubowski, 2007). In order to do this we, as teachers, have to learn how to think creatively while problem solving.

Theoretical Framework.

According to Guilford (1967), divergent thinking is the mental process by which we draw from many ideas while searching for different solutions to the same problem or set of situations. In this study, creativity is defined as “the process of sensing gaps and disturbing missing elements; forming ideas or hypothesis concerning them; testing these hypotheses; and communicating the results; possibly modifying and retesting the hypothesis again” (Torrance, 1962, p. 16). According to Guilford and Merrifield (1960) there are six kinds of thinking abilities that involve creativity and divergent thinking (shown in Figure 1).


Figure 1. Six kinds of creative thinking abilities.

Purpose.

The purpose of this on-going study is to examine practicing middle and secondary teachers’ different strategies and methods of solving given problems in a hybrid geometry course. The authors will address the following research questions:

1. In what ways, if any, do in-service mathematics teachers demonstrate divergent thinking and creativity skills while problem solving?

2. When teachers spend time inventing new solutions, what kind of thinking abilities as defined by Guilford and Merrifield (1960) emerge?

Setting.

The geometry course used in this study is designed to encourage divergent thinking, creativity, and reflection during small group problem-solving with a hybrid (online and face-to-face) environment. According to Cropley (2001), the environment must include resources to make creativity available and rewards for those who diverge from the usual. In this course, teachers typically explore problems and seek multiple solutions while using technologies such as graphic calculators, data collectors, dynamic geometry software, spreadsheets, and online resources. By reflecting on and discussing multiple solution methods for given problem sets, it is expected that teachers will become more aware of their own abilities to think divergently and creatively. The participants were 25 middle and secondary mathematics teachers enrolled in a graduate level geometry course as part of a teacher education Master’s Degree program at a regional university in the southeastern United States. The teachers were randomly divided into six problem-solving groups for the duration of the course.

Methodology and Data Collection.

This research followed to model of case study set forth by Creswell (2003). Creswell (2003) defines a case study as an exploration of a 'bounded system' or a case over a period of time through detailed, in-depth data collection involving multiple sources of information rich in context. Using multiple sources of qualitative data - online discussion board entries, observations of in-class activities, and written solutions to given problems along with students' reflections on these same problems - case study analyses were undertaken to identify elements of divergent thinking and creativity used while problem solving.

The Assignment.

Divergent thinking can be assessed using divergent thinking tasks. Divergent thinking tasks are intended to capture the creative quality of the participants’ responses, not simply the number of responses (Silva et al, 2008).  In this study, teachers were instructed to find as many different processes/procedures as possible when solving each problem. In this paper, the solutions of a group of teachers are shared and discussed.

The Problem.

Two brothers are part of an expedition and they have discovered a new island. From the sky they notice that the island is shaped like an irregular convex quadrilateral. They are not sure of the actual dimensions of the island, but they want to determine a way that they can fairly divide the island between the two of them.

The class determined the following to be features of the problem:

    • The island is shaped like an irregular convex quadrilateral.
    • The island dimensions are unknown.
    • Each brother’s portion of the island must have an area equal to the other brother’s portion.

Teachers were given one week to work with their groups and submit their solutions electronically online. Analysis of the groups’ solutions revealed that the three thinking abilities displayed were Fluency, Flexibility, and Originality while solving this problem.

Fluency (F): Shively (2011) states the first step in problem solving is to have as many ideas as possible to choose from and “play with” if the endeavor is to be creative. Fluency, as defined by Guilford and Merrifield (1960) is the ability to generate a lot of ideas which increases the chances of significant responses or in the case of mathematical problem solving, meaningful or correct solutions. Conway (1999) suggests one way to determine Fluency while problem solving is to simply count the number of correct solutions.

Flexibility (X): Flexibility, according to Shively (2011), is the ability to look at a problem from many different angles. “The measurement of flexibility considers how many different ways of thinking about a problem are employed” during the problem solving process (Conway, 1999, p 511). Flexible thinkers discover new possibilities, employ different interpretations, and are able to shift from one approach to another.

Originality (O): Originality refers to solutions that are unusual, infrequent, insightful, or clever. Original thinkers produce unexpected ideas, first-of-a-kind solutions, and have the ability to think beyond the obvious (Conway, 1999; Shively, 2011; Torrence, 1972). “Originality requires the greatest risk-taking and is the crux of innovation, yet it is the most fragile dimension of creativity in school settings oriented to correct answers” (Shively, 2011, p. 12).

Solutions.

This geometry problem resulted in total of 4 strategies and 7 different methods for the entire class. For Group A, the problem resulted in a total of 3 strategies and 5 different methods as seen below.

Strategy 1: “Cake Cutting”: Figure 2 shows Group A’s initial analysis of the problem. “The first suggested method is to divide the land by simply drawing a diagonal.  We found that this would not be fair. We are given that the island is an irregular convex quadrilateral with unknown dimensions; therefore we cannot be sure that this method will produce equal parcels. In fact, an equal division in this manner would be an unlikely coincidence. However, even if with an uneven division using this method, a fair result could be achieved if the brothers agreed that one brother choose the direction of the division, then the other brother claim a parcel. This is the ‘cake cutting’ method that depends on the goodwill of the brothers, so it is not optimal.” This method demonstrates the group showed risk-taking and offered a clever, unexpected result.  They gave no thought to finding the actual answer, but rather a creative possibility.  This shows Originality.


Figure 2. Sketch of cake-cutting strategy.

Strategy 2: Midpoints of the Sides of the Quadrilateral: After much debate, the teachers in Group A decided to use the dynamic software program Geometer’s Sketchpad to divide the land by connecting the midpoints of each side to the midpoints of its adjacent sides. The resulting quadrilateral is a parallelogram whose area is equal to half of the total area. This method showed flexibility because the group approached the problem from a different perspective. Their sketch is shown in Figure 3.


Figure 3. Quadrilateral midpoint strategy - A.

While correct, not all of the members of Group A were satisfied with this solution. One female student stated, “We can give the interior quadrilateral to one brother and the remaining 4 triangles to the other brother. However, although this resolves the problem of dividing the island evenly, it is not quite fair. One brother would acquire all of the beachfront property whereas the other brother would acquire none.” The idea of fairness was very important to this student and led her to pursue another solution.

Another attempt at a solution from Group A started with marking the midpoints on each of the sides of the quadrilateral, then the midpoints were connected with the opposite midpoint to form line segments. The intersection of these segments was marked. These line segments formed 4 quadrilaterals within the original quadrilateral. This group found that the sum of the areas of the non-adjacent quadrilateral is equal to the sum of the remaining quadrilaterals. This strategy is shown in Figure 4.


Figure 4. Quadrilateral midpoint strategy - B.

Strategy 3: Triangle Areas: The following solution also used the midpoints of the sides of the quadrilateral, but the groups focused on their prior knowledge of triangle areas instead of the area of quadrilaterals. In an attempt to solve this problem, Group A decided to construct the line HE that joins the midpoints of side AD and side BC respectively, then the line FG that joins the midpoints of sides DC and AB. They labeled the intersection point as X. Next, they constructed the diagonals from X to each of the vertex angles to form triangles. These first steps are shown in Figure 5.


Figure 5. Triangle area strategy - A.

They shaded the triangular areas of AXB, XCB, XCD, and ADX and used Geometer’s Sketchpad to measure the triangular areas. This next set of steps is shown in Figure 6.


Figure 6. Final steps in triangle area strategy - A.

Group A gave an alternate method of finding the Triangle Area. They constructed midpoints and drew a line segment between the midpoints of the opposite sides of the quadrilateral. Next they constructed line segments from the vertices to the intersection of the midpoint connector lines to find the centroid of the quadrilateral. They described their process as follows: “Disregarding the midpoint connector lines, we could see four triangles.  We knew that the medians we constructed between the midpoint of the hypotenuse and the opposite vertex would divide each of the four triangles into two equal parts (ex: CID is divided into GIC and DIG). If we assign half of each triangle to a brother, they will both receive the same amount of land. We decided to group the eight triangles into four quadrilaterals (the quadrilaterals are HAEI, EBFI, FCGI, and GDHI) so the brothers will each have two sections of land. Each brother will have opposite quadrilaterals, HAEI and FCGI are for one brother and EBFI and GDHI are for the other brother.” This work is shown in Figure 7.


Figure 7. Triangle area strategy - B.

Strategy 4: Triangle Altitudes: This strategy did not involve the midpoints of the quadrilateral sides, but rather focused on the diagonals of the quadrilateral, shown in Figure 8. The group found the midpoint, point I, of this diagonal segment to create equal bases. They also constructed a segment from the midpoint of the diagonal to the opposite vertices. In doing so, they constructed two triangles (with equal bases and equal heights). Therefore, GIH and JIH have the same area. Likewise, GIF and FIJ also have the same area. They shaded this in the diagram and also measured the areas in Geometer’s sketchpad to show equality.


Figure 8. Triangle altitude strategy - A.

Another group used the idea of triangle altitudes along with the midpoint of only two sides of the quadrilateral. “We begin solving this problem by first using Sketchpad to construct the midpoints of the two opposite sides C͞B and A͞D. After finding the midpoints of the two opposite sides C͞B and A͞D, we construct D͞E, F͞B, and D͞B to form 4 triangles, DCE, DEB, ABF and DBF. Next we construct GD, the height or altitude for DCE and DEB. We call it h1. The next step is to construct B͞H, the height or altitude for ABF and DBF. We call it h2. These heights are used to determine the areas of the respective triangles (see Figure 9). One brother will get the outer triangles; the other brother will get the inner quadrilateral.

 
Figure 9. Triangle altitude strategy - B.
 
Figure 10. Clean sketch of triangle altitude strategy - B.

This geometry problem resulted in a total of 4 strategies and 7 different methods for the entire class, showing flexibility for the class as a whole. The majority of the groups created triangles and focused on triangle areas. Another popular strategy was to use the midpoints of the quadrilateral in their methods. The idea of fairness forced the class to find multiple ways of dividing the land to be the “most fair” which was evidence of the fluency.  The “cake cutting” method showed originality. Group A was the only group to demonstrate fluency, flexibility, and originality. Upon analysis of the student work, there was no evidence of sensitivity, elaboration, or redefinition, possibly due to the type of problem and juxtaposition of the problem to the course work. It was assigned after a lesson on quadrilaterals so the students immediately converged to the idea of using midsegments.

Conclusions.

The results of this problem are just one example of the levels of divergent thinking skills that emerged during this study. The in-service teachers in the course were assigned a set of open-ended problems and asked to solve them. They were later specifically instructed to “think outside the box” and find as many solutions as possible for each problem. In-service teachers are engaging in divergent thinking and seeking creative ways to solve geometry problems, however, this only materialized after they were specifically instructed to do so. Once specifically instructed to find many solutions, teachers working in groups demonstrated some areas of the Guilford Model for divergent thinking. When teachers were not specifically instructed to seek many solutions, some groups quickly reduced to one method of solution and stopped. If we teachers value divergent thinking, then we can foster an environment of divergent thinkers in our classrooms. Engaging ourselves in creative problem solving activities may be a first step.

The use of technologies such as MS Excel and Geometer’s Sketchpad served as key elements in sparking divergent thinking and mathematical creativity. The technology assisted the teachers’ exploration of ideas and verification of possible solutions, and also helped to dispense with erroneous plans or solution methods. The technology promoted creativity in Group A. It was also observed that the teachers became more confident when discussing their ideas online rather than in class. They developed “keyboard courage” that they typically do not have in a face-to-face setting. The online environment allowed a better sense of equitable footing in the group and transparency amongst group members that helped to form a comfort level with teachers. 

References

Conway, K. ( 1999). Assessing open-ended problems. Mathematics Teaching in the Middle School, 4(8), 510-515.

Corey, D., Unal, H., Jakubowski, E., An 8th grade Geometry Problem from Japan and American Teachers Solutions, Learning and Teaching Mathematics, Vol. 5, Issue 5, pp. 12-16, (2007)

Corey, D. & Unal, H. (2004). Activating Metacognition via Distance Learning: An Exploratory Study of In-Service Math Teachers’ use of Metacognitive Monitoring Skills in an Online Problem Solving Course. In C. Crawford et al. (Eds.), Proceedings of Society for Information Technology and Teacher Education International Conference 2004 (pp. 4385-4388). Chesapeake, VA: AACE.

Cropley, A.J. (2001). Creativity in Education and Learning: A Guide for Teachers and Educators. London: Kogan Page Limited.

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Guilford, J.P. & Merrifield, P.R. (1960). The Structure and Intellect Model: Its Uses and Implications. Los Angeles: University of Southern California.

Kwon, O.N., Park, J.S., & Park, J. H. (2006). Cultivating Divergent Thinking in Mathematics through an Open-Ended Approach. Asia Pacific Education Review 2006, 7(1), 51-61.

National Council of Teachers of Mathematics. (2009). Principles and standards for school mathematics. Reston, VA: Author.

Shively, C. H. (2011). Grow creativity. Learning & Leading with Technology, 38(7), 10-15.

Silva, P., Winterstein, B.P., Willse, J.T., Barona, C.M., Cram, J.T., Hess, K.I., Martinez, J.L., and Richard, C.A. (2008).  Assessing creativity with divergent thinking tasks: Exploring reliability and validity of new subjective scoring methods.  Psychology of Aesthetics, Creativity and the Arts, 2(2), 68-85.

Torrance, E. P. (1962). Guiding creative talent. Englewood Cliffs, NJ: Prentice Hall.

Treffinger, D.J., & Isaksen, S.G. (1992). Creative problem solving: An introduction. Sarasota, FL: Center for Creative Learning.

 
Nikita Patterson
Georgia State University
npatterson@gsu.edu
 
Darryl L. Corey
Radford University
dcorey3@radford.edu

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