Using Problem-based Learning Activities to Identify Creatively Gifted Mathematics Students

Scott Chamberlin
University of Wyoming, USA

Abstract

This chapter investigates the potential of problem-based learning activities as a vehicle for assessing the creativity of gifted mathematics students. Presented with the problem scenario of a physical education teacher who wants to know if one can accurately predict from body size which sports event a student would excel in, a group of 36 grade 4 and 5 students was asked to analyze relevant data available on the Web to determine whether a relationship exists between body size and prowess in specific athletic events. Samples of typical and potentially creative student responses are presented, and the implications for using such tasks to assess creativity are considered. Findings suggest that problem-based learning tasks can lend themselves to the identification of creatively gifted mathematicians.

Introduction

Problem-based learning (PBL) has been adopted at all levels, from kindergarten through graduate school (Hmelo-Silver, 2004; Tan, 2005),

Special thanks to Bob Simpson, district mathematics coordinator, and Darcie Achord, grade 5 teacher, for arranging the problem-solving opportunity and for facilitating the problem with grade 4 and 5 students.

and this is made possible by the flexible nature of its core features. The core features are the same regardless of the grade level, but adaptations are needed to accommodate age-related ability levels. These features include the emphasis on self-directed learning, collaborative problem solving, as well as open-ended tasks that are set in realistic contexts, are interdisciplinary in nature, and foster higher-order thinking (Chin & Chia, 2004; Cockrell et al., 2000; Dunlap, 2005; Goodnough, 2003; Hmelo-Silver, 2004; Majeski & Stover, 2005; Nelson et al., 2004; Savin-Baden & Wilkie, 2004; Tan, 2005; Van Liet, 2005). Students engaging in PBL tasks go through several steps: meeting the problem, defining the problem, gathering facts about the problem, hypothesizing solutions to the problem, researching the problem, rephrasing the problem, generating alternative solutions, and advocating solutions to the problem (Fogarty, 1997).

This chapter addresses the use of PBL activities as a curricular tool to identify students who are creatively gifted in mathematics by analyzing their responses to a problem. Following a brief review of the literature, the educational context for administering the problem is provided. Subsequently, sample student responses are discussed in an attempt to provide an understanding of the differences between typical and creative responses. The chapter concludes with a discussion of assessment practices.

Mathematical Creativity

Although mathematical creativity, giftedness, and PBL have been discussed in the literature, empirical studies on the interrelationships between these three areas do not exist. Consequently, this literature review will merge several areas of research to provide a background for the present discussion.

In modern history, Balka and Krutetskii brought attention to the question of creativity in mathematics. Balka's dissertation (1974a) reviewed existing instruments for measuring creativity and propelled the discussion into the arena of mathematics education in the United States. In the same year, he published an oft-cited article that detailed the criteria used to characterize creativity in mathematics (Balka, 1974b). Similar to Balka's work, Krutetskii's research (1976) focused on several aspects of creativity, and he suggested that the identification of mathematical creativity must be within the setting of mathematical problem solving. According to many researchers, a critical component of creativity is flexibility in thinking (Helson & Crutchfield, 1970; Keisswetter, 1983; Krutetskii, 1976; Silver, 1994). Krutetskii was the first to equate mathematical creativity with mathematical giftedness, and others have since made the same claim (Haylock, 1997), although Silver does not agree with that (personal communication, February 2, 2007).

Over ten years ago, Zentralblatt für Didaktik der Mathematik (ZDM), a leading international journal in mathematics education, dedicated a special issue to creativity in mathematics. In this issue, Ching (1997) discusses types of problems that can be used to assess creativity in students. This discussion is similar to the focus of this chapter in that it explicates assessment tools that may be employed to identify creatively gifted mathematicians. Silver (1997), on the other hand, examines mathematical creativity and how it can be developed through mathematical problem solving and problem posing. He identifies the three components of mathematical creativity as fluency, flexibility, and novelty. This conception of creativity is similar to that of Torrance (1974). Leung (1997) also looks at problem posing as an indicator of mathematical creativity. His argument is that problem posing can foster mathematical creativity by getting students to think flexibly. More specifically, Leung agrees with Mamona-Downs (1993) and Van den Heuvel-Panhuizen and colleagues (1995) that "problem posing means the formation of novel problems, with solutions unknown to at least the one who formulates it" (p. 81). Similarly, Pehkonen (1997) discusses the value of mathematical creativity and what can be done in the classroom to foster it. It seems logical that the pedagogical approach and the curriculum are the two factors that would make the greatest impact on developing creativity in the classroom. This phenomenon has yet to be researched, although Silver (1997) suggests that open-ended, ill-structured tasks are best at promoting creativity in mathematics.

Following ZDM's special issue, Krulik and Rudnick (1999) considered teaching techniques for improving creative thinking in mathematics. One of their suggestions was that Polya's (1973) fourth step to problem solving be changed from "look back" to "reflect." In so doing, students look critically at their answers. Ideally, reflecting will encourage students to seek additional solutions to problems rather than just accepting the first "standard" solution to a problem. The emphasis becomes finding as many solutions as possible and not merely identifying the one obvious solution. Hence, thinking in flexible ways is fostered.

Perhaps the article most germane to this chapter is that by Williams (2002), which discusses a tool for assessing creativity in the classroom. Specifically, utilizing Bloom's taxonomy (Bloom et al., 1956), he created indicators of cognitive processes that may lead to a creative product and designed the tool based on five seminal articles on mathematical creativity (Chick, 1998; Dreyfus et al., 2001; Dreyfus & Tsamir, 2001; Krutetskii, 1976; Williams, 2001).

An article that has had a significant influence on the field of gifted education is "Creativity: The Essence of Mathematics," in which Mann (2006) describes myriad definitions of mathematical creativity as they relate to mathematical giftedness and considers implications for instruction and assessment. E. Paul Torrance (1974) has perhaps made the biggest impact on mathematical creativity in gifted education. His test of creativity, according to Haylock (1997), employs indicators of mathematical creativity that comprise fluency, flexibility, and originality. Fluency is measured by the number of acceptable responses, flexibility by the number of different kinds of responses, and originality by how often specific responses are produced by a respondent relative to how often the group provides such responses. Novelty or originality is perhaps the gauge of creativity most often employed in the educational world. Consequently, novelty in mathematical solutions has been used in this chapter to refer to mathematical creativity. Adapting Marland's (1972) definition, giftedness is defined here as any high performance with demonstrated achievement and/or potential ability in creativity or productive thinking. As is apparent in the literature review, there is no one agreed method for assessing mathematical creativity. For this study, PBL activities were chosen as a vehicle to assess and identify mathematically creative students.

The PBL Task

A mathematical PBL activity called the Athletics Problem (see Appendix) was implemented over a two-week period in the Rocky Mountain region of the United States. The participants in this exploratory study had not previously been identified as gifted, although they were picked out by their teachers as top students in general education. The problem required the use of the statistical concepts of standard deviation (SD) and correlation, specifically because they had not been introduced to the students in their mathematics curriculum. Hence, novelty was a major criterion in selecting the mathematical concept.

The task was on athletics (track and field) and consisted of a scenario in which an elementary physical education teacher is trying to ascertain whether or not body size, specifically height and weight, has any relationship to the optimal athletic event for an athlete. Statistical terms, such as SD and correlation, were not mentioned in any part of the problem so that students would not be prompted to apply statistical processes in the event that they were aware of them or found them in a resource. Moreover, students were given only one week to complete the task, and any use of statistical terminology may provide clues regarding how to efficiently solve the task. Web sites are available that detail athletes' heights, weights, and the athletic events in which they competed (e.g., U.S.A. Track and Field Athlete Bios on http://www.usatf.org and Indiana Invaders Athlete Bios on http://www.indianainvaders.com,). The USATF site is the most comprehensive source available, as it provides both women's and men's data. Students were asked to relate an athlete's best event with body size, for either women or men. In essence, this task demanded that students create a model to look at body size data (variable A) and decide what event is best for an athlete (variable B).

Method

Students were recruited by the mathematics coordinator in one of the largest districts in a Rocky Mountain state. To identify students, the coordinator asked all grade 4 and 5 teachers to identify no more than three of the most advanced mathematicians in their classes. This was done in an attempt to identify students who were capable of using high-level mathematics on a challenging problem. The selected students met at an elementary school on April 24 and May 1, 2007, from 9:30 a.m. until 11:00 a.m. In the first meeting, they were put into groups according to schools, so as to allow students the opportunity to work on the task in groups when they returned to their own school. Following an explication of the problem and a brief discussion about track and field events, students were instructed to work on the task in the time remaining that day and then as homework, and they were told that it was imperative that each group understand all of the mathematics in the problem such that any one of them could provide an explanation and rationale to the large group upon returning the next week. Subsequently, the author and two facilitators went round the room to ensure that students understood what was asked in the problem statement. One week later, students assembled to present their problem solutions. Their work was collected and analyzed to identify whether responses were typical of students of this age or whether potentially novel solutions existed, using a four-level rating system designed by Livne, Livne, and Milgram (1999). This rating system places a large emphasis on novelty. The four levels of understanding are, briefly, (1) ordinary: initial impression from data on the surface, (2) mild: attention to details, (3) moderate: integration, and (4) profound: transfer/application.

Results

Solutions that are sound mathematical responses but not creative ones are discussed first, followed by two responses that appear to warrant further consideration for evidence of creativity. It is important to note that the responses that are not creative are not poor responses. In general, students answered the given question thoroughly and used mathematics that was advanced for grades 4 and 5. Novel responses were sought in an attempt to identify potential creativity.

The approach most frequently used, by five out of the total eleven groups, was to look at the range of body sizes and graph the data in an attempt to predict any correlation between body size and athletic event. Another typical approach, employed by four groups, was to use some or all measures of central tendency, such as mean, median, and mode. Rudimentary versions of SD and correlation were used to analyze the data spread. Range is a simplified version of SD; and although it is somewhat sophisticated for students of this age, it is not novel per se. All groups made graphs or charts of the data in an attempt to represent them pictorially. Graphing is a large part of curricula in grades 4 and 5 in the United States. Table 10.1 is a typical response from a group. These nine typical responses were rated 1.

Of all the groups, two groups appear to have come up with potentially creative responses with a rating of 3. One of these groups responded that it is possible to predict the optimal event for some athletes but not for others. This response may be considered creative partly because of the

way the question was phrased: "Using data from a Web site, is it possible to look at an athlete's height and weight, without knowing what event they do, and predict the best event for the athlete?" It is likely that the typical elementary student would view such a question as entirely dichotomous. That is to say, given the understanding that most elementary students have of SD and correlation, this question would be interpreted as the existence or not of a relationship. Practically speaking, to upper elementary students, body size does or does not indicate a propensity for success in a certain track and field event. As none of these students had been introduced to the concept of SD or correlation prior to the activity, this response is novel and may deserve consideration as a potentially creative response. Students at this age and with this level of knowledge are likely to view the response as one of the two extremes.

This group deduced from their analysis of data from usatf.org that body size can indicate what event in which track (running) athletes will do best. However, their analysis suggested that size does not indicate what event in which field athletes will do best. Their spread of data was minimal for running events (low SD) and scattered or large for field events (high SD). They stated that distance athletes are typically slender and short to medium in height. Middle-distance runners are also slender and typically of medium height. Sprinters are often of medium height and heavier than distance and middle-distance runners, because of the need for upper body strength. However, field athletes come in many different shapes and sizes. In essence, this group suggested a potentially creative response to this problem by looking at all events individually. They may have answered the problem in much the same way a highly trained statistician would. They stated that, for track athletes, accurate predictions could be made about success in athletic events. However, for field athletes, only weak predictions could be made from the data. Moreover, this group read into the problem such factors as muscle type and training. They sought more detailed information about such factors as leg height, upper body strength, and so on.

The other group created a mathematical model to answer the question, with a matrix to classify athletes as tall, medium, or short (in height) and heavy, moderate, or light (in weight). A sample matrix is recreated as Table 10.2. With this matrix, along with parameters defining each of the nine categories of height and weight, the students simply put athletes into categories to ascertain whether a large or small spread existed. The more categories that are represented by athletes from the same event, the larger the spread of body sizes. In turn, a large spread indicates that it might be difficult to predict success in a certain athletic event. For instance, if the decathlon has athletes in four boxes, perhaps tall moderate, medium moderate, tall light, and medium light, then the spread may be considered large, indicating a low correlation. On the contrary, if the 10,000 meters has athletes in two boxes, such as medium light and short light, then the event may be regarded as having a smaller spread, suggesting a high positive correlation. Data characterized by a small spread can be used to more closely predict future success in an athletic event.

Methods of Assessment

The focus of this chapter is to use PBL tasks as an assessment tool to identify creatively gifted mathematicians. An important caveat is that measuring creativity with only one analysis of a PBL task is problematic. The National Association for Gifted Children (2005) recommends using multiple pieces of data to make conclusions during assessment in identifying giftedness. Similarly, with creativity, multiple pieces of data must be analyzed and assessed before any suggestion of creativity can be taken seriously.

Two factors need to be taken into consideration when designing a PBL task to identify creatively gifted mathematicians. First, the task must have sufficient depth to precipitate multiple, creative responses as the Athletics Problem did. Second, working on the presupposition that the task is an adequate one, the teacher must have some method of assessment to identify creative responses.

Two methods of assessment can be utilized to track highly creative responses over time. One method tracks students' performance on one problem-solving task, while the other tracks students' performance on a series of problem-solving tasks. The first method entails keeping an electronic database of responses to allow comparison of current with previous responses. The second method involves having students work in various groups over the course of a school year and tracking the level of creativity in each individual.

The first method requires the accumulation of responses from numerous students in the electronic database. After years of administering the same task, a pattern of common responses will surface. When a new or novel response is elicited after years of administering the activity, it is likely that the response suggests student propensity for creativity. One way to expedite this process is to have several teachers in a consortium or a central group, such as a mathematics teachers association, administer the problem so that a large database can be attained in a relatively short period of time. Ten teachers administering a problem to three different classes with seven groups in each will produce 210 responses in a matter of weeks. A period of 10 to 12 years may be required for a single teacher to attain 210 responses. With numerous responses gathered by multiple teachers, it is easy to see recurring responses among them and identify novel or less common responses relative to frequently occurring ones. In the case of the Athletics Problem, along with the rating system, knowledge of curriculum standards and intuition developed after years of experience were relied upon to identify creative responses. It would have been easier to quantitatively document creative responses among an abundance of responses in an electronic database than it was with a mere 36 students each doing the same problem once.

The second method is a more complicated process than the first, but it too can yield valuable information regarding creativity. The difference between them is that this method tracks students over the course of a school year whereas the first tracks students on only one performance. As it is imperative to have multiple pieces of data for consideration of creativity, this method may be more valid than the first one because it is not a one-time analysis. In this method, individuals in each group receive mean scores and by midyear creative individuals may be identified quantitatively. Therefore, if the groups in which student A is working score high continually on a creativity scale, then student A probably has a propensity for creativity. An example will clarify how this method works. In a class of 20 students where each student is identified by a letter (A through T), completed PBL tasks are turned in to the teacher throughout the course of the year with students' identification letters written on the back of the work to indicate ownership. If students A, B, and C did PBL task 1, then the group's mean creativity score received for the task is recorded under students A, B, and C. For the next PBL task, if students A, D, and E work together, the mean creativity score will be recorded under students A, D, and E. Ultimately, the progression of scores will follow each student, regardless of the student's group. That is to say, if student B has continually produced less than creative work, it would show after a series of PBL tasks done with various groups, even when student B has worked with a creative individual. Similarly, if student A is highly creative in general, it should theoretically show repeatedly, even when the student has worked with less than creative peers. By having a rating system that focuses on creativity and anonymous grading of work, an evaluator can be confident that creativity exists, or does not, because a pattern has been created.

In the present study, assessment was guided by three components: ratings, questions, and cognitive tasks. Rating was guided by the key mathematical terms used in the cognitive tasks, which were group presentations and written work. The instructors asked questions to ascertain the mathematical processes applied by students. The three components work in combination and are difficult to separate during analysis. In many cases, students did not know advanced mathematical terminology, so the instructors looked for grade-level terminology. As an example, students in grades 4 and 5 often use average to represent any part or some combination of central tendency. Therefore, when the word "average" was mentioned, the instructors would ask students the process they had used to derive the average and, from the students' elaboration, determine if it was mean, median, or mode. In addition to questions, there were individual group presentations accompanied by written explications of processes, so the instructors were able to match their notes with students' written work.

Conclusion

Although algorithms are efficient ways of solving complex problems, their use by students may not indicate mathematical creativity (Haylock, 1997). Students can be expected to use algorithms for simplistic calculations, but the mathematically creative individual may not. Another tactic that can be deployed to gauge mathematical creativity is to have students recreate the problem or pose it in another way (Silver, 1997). Having students create their own tasks is a way to make the problem meaningful to them. One caveat with this approach is that the teacher has to use the same problems if employing the first method discussed above to evaluate mathematical creativity. If the teacher keeps adopting new problems, no pattern of typical responses may be established.

In addition to their use as a curricular tool to identify mathematically creative students, PBL tasks may hold promise for fostering creativity. Silver (1997) writes, "The development of students' creative fluency is also likely to be encouraged through the classroom use of ill-structured, open-ended problems that are stated in a manner that permits the generation of multiple specific goals and possibly multiple correct solutions, depending on one's interpretation" (p. 77). He did not make this statement in reference to PBL tasks. However, his description of problems should seem strangely familiar to users of PBL. Characteristics such as "ill-structured," "open-ended," "multiple specific goals," and "multiple correct solutions" should encourage PBL users that PBL tasks may lend themselves to creativity development. Students in PBL are not asked to select the one method for solution and implement it. Instead, they are required to identify a solution, implement the solution, and then determine whether the solution meets the needs of the problem. It is important to note that students rarely select a creative option overtly. Rather, they simply create a solution, implement it, and then see if it works.

A caveat here is that the most obvious of the three components of creativity has been used in this chapter, the one with which most readers identify: novelty. However, Torrance (1974) and Silver (1997) suggest that fluency and flexibility are also components of creativity. Fluency, according to Torrance and Silver, is reflected in the number of responses that can be generated to a question. Flexibility, as the term suggests, is shown by the varied responses produced and the ability of the solver to work from more than one perspective. Having students present problem solutions to other groups also promotes flexibility because they are made to examine the problem from more than one perspective.

It is important to note that mathematical creativity may surface in multiple modes, and standardized test data often do not reveal such information (Balka, 1974a). Other mathematical problem-solving approaches have been used to identify creatively gifted mathematicians, such as model-eliciting activities (Chamberlin & Moon, 2005).

Appendix The Athletics Problem

Summer is on the horizon and tomorrow is the last day of school at JC Watts Elementary. The last day of school means one thing to all students—field day. Each year, students look forward to competing against peers one year older or younger than they are. There are three separate age-group categories for Watts Field Day: Kindergarten/1st, 2nd/3rd, and 4th/5th. Field day is an entire day in which students take part in several competitions for fun. Younger students, of grades K–1 and 2–3, compete in activities such as blowing up a balloon, eating blueberry pie, tug of war, and egg toss. Older students, of grades 4–5, compete in track and field events, such as the long jump, shot put, hurdles, high jump, 100 meters, 200 meters, 400 meters, 800 meters, the mile, and the two mile.

The Watt's physical education teacher, Mr. Finley, has students practice these track and field events to prepare for field day. He thinks that students with certain body sizes tend to do well in certain events. As an example, it has been his experience that tall students often do better at the high jump than shorter students. Light-weight students often fare well in the endurance or distance events like the mile and the two mile.

Mr. Finley is seeking some sort of method to determine whether or not he can look at a student and decide the optimal event for the student. As an example, can Mr. Finley simply look at a student and say "You should be a good shot putter and you should be a good twomiler" without even seeing the student perform any events? He has been perplexed by this thought for some time. He has actually collected some informal data. However, he feels that data on current and future U.S. Olympic athletes, such as data from USATF, United States of America Track and Field, an organization that governs and funds Olympic athletes throughout the United States, would be more accurate than his data.

Using data from a Web site (e.g., http://www.usatf.org/athletes/bios/ or http://www.indianainvaders.com/athletes/index.asp), is it possible to look at an athlete's height and weight, without knowing what event they do, and predict the best event for the athlete? To select an athlete, simply click on the name, and the vital statistics, such as event, weight, and height, will appear.

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