Abstract
The purpose of this exploratory study was to assess the effect of a novel approach to mathematics instruction on gifted high school students’ engagement, motivation, and metacognition. Participants in this study included gifted students who were enrolled in a 3-year, residential, specialized mathematics and science high school. Rather than respond only to problem sets assigned by the teacher, students were asked to create original, conceptual mathematics problems at several points over an academic semester. After completing the assignment, students were administered a follow-up Likert-type scale and open-ended survey designed to assess sources of motivation and rationales for choosing a particular type of problem. No differences were found between gender or grade, but patterns of engagement, challenge, and intrinsic motivation were found in open-ended responses.
Consider the following problem, composed (and subsequently solved) by a gifted high school sophomore for an assignment involving the writing of an original problem in a precalculus course:
The human eye is very unique, and it shapes our perception of the world around us. But why do we perceive things as we do, and how is this linked to mathematics? This is the question I set out to explore with this original problem. In everyday life, objects appear smaller, larger, thinner, thicker, and a host of other alterations depending on how we view them. I chose to study how the size of an object is affect[ed] by moving away or towards it. Looking online, I found a study to find the optimized angle, but I still wondered how mathematicians arrived at this point. The problem I am about to present is my attempt at blending reality and the world of mathematics to create a problem. A man observes a painting hung on a wall from several feet away, looking directly at it. He realizes that the viewing angle changes as he moves away from the painting, getting larger until it hits a point of optimization, after which it begins getting smaller. However the man cannot accurately conclude that this supposed point exists where the angle is at its maximum without mathematics. Write a proof that proves the optimization to have a larger angle than all other possible points given the images below.
What kinds of instructional strategies effectively engage and sustain motivation among students who are gifted and talented in mathematics (or another domain)? Are students differently or more deeply engaged in mathematics when they are asked to create problems rather than answer them? Matsko (2011) presented a novel approach to and reflection on developing creative thinking skills and deepening conceptual understanding by asking gifted students in mathematics courses to create their own problems, like the “point of optimization problem” above:
Anyone can write tedious, difficult problems that review core math subjects, but to write problems in a novel, challenging, and refreshing manner, one must be imaginative. I feel that this creative side of math is an often overlooked aspect of the field as many believe math to be an extremely black-and-white, rigid, and boring subject. (p. 1)
In this study, we surveyed and examined the assignments of students enrolled in a specialized high school for students who are gifted and talented in mathematics and science. With Matsko’s (2011) experiences with students enrolled in Advanced Problem Solving (APS) in mind, we wondered,
whether the opportunity to create original mathematics problems would enhance levels of challenge, creativity, and intrinsic motivation; and
whether first-semester sophomores in their problem-creation exercise would differ from upperclassmen in the same exercise.
Prior to developing this study, the exercise was piloted and refined with students in upper level mathematics courses such as APS and Advanced Placement BC Calculus.
Conceptual Framework
Silver (1997), in examining multiple approaches to the instruction of mathematics, provided a departure point for this study:
Mathematics as an intellectual domain stands at or near the top of any hierarchical list of intellectual domains ordered according to the extent to which creativity is evident in disciplinary activity or production. Thus, it is ironic that for most students throughout the world, mathematics would almost certainly be among the set of school subjects least associated with creativity. (p. 75)
Silver (1997) suggested that creativity is a disposition that is underdeveloped and underappreciated in mathematics instruction and that inquiry-based mathematics (including problem posing and problem solving) can effectively develop “deep, flexible knowledge in content domains” (p. 75). There is indeed a paucity of empirical literature that explores creativity and mathematics and specifically links motivation to creativity. Before exploring the literature in these areas, however, it is important to place this study in the conceptual framework of motivation theory.
Motivation in mathematics is of particular concern for educators in both gifted and general populations. There is evidence of general decline in motivation in mathematics from Grade 9 through Grade 11 (Chouinard & Roy, 2008). According to Preckel, Goetz, Pekrun, and Kleine (2008), while, in the general population, mean differences in mathematics ability are relatively small, “males show higher mathematics competence beliefs, a stronger interest in math, and a stronger performance orientation in mathematics than do females” (p. 149). With declines in motivation and gender-based motivational differences in mind, can math educators, as Silver (1997) suggested, enhance motivation through innovative learning opportunities?
Very broadly, this study is anchored in the concept of intrinsic motivation, which “refers to motivation to engage in an activity for its own sake” (Pintrich & Schunk, 2002, p. 245). In other words, given an array of options to act upon, what do we choose to do because of our enjoyment of the activity?
Motivation may be understood as a process that begins with initial engagement and moves toward sustained engagement and self-regulated strategies. Therefore, this study is specifically characterized by dimensions of self-determination theory of motivation, as developed and explored by Deci and Ryan (2000, 2002). Self-determination theory suggests that (a) individuals are aware of their needs, their strengths, and their weaknesses; and (b) decisions to act and satisfy needs depend on our understanding of them.
In particular, we believe that in the problem-creation exercises in this study, there will be evidence of two of the psychological needs/dimensions present in the decisions that direct behavior in self-determination theory, namely, autonomy and competence. Feelings of competence are necessary because we must feel we have an ability to master both our environment and the multitude of interactions in our lives. Second, humans have a common and innate need to act autonomously and with a certain degree of control over their lives.
With respect to the mathematics problem-creation exercises explored in this study, we were interested in the ways in which and degrees to which students’ experiences enhance intrinsic motivation by
presenting a challenging activity, which enhances self-efficacy and feelings of competence;
engaging curiosity about problems that seem complex or incongruous; and
allowing students a sense of control and ownership over their own learning.
Lepper and Hodell (1989) asserted that these dimensions are integral to enhancing intrinsic motivation. A fourth dimension, fantasy, is not being explored in this study.
In a study of mathematics, Banda, Matuszny, and Therrien (2009) suggested that engaging students in developing higher order mathematics skills consists of presenting mathematics tasks that students prefer to enhance subsequent interest in solving difficult math tasks. To this point, learning in a nontraditional (i.e., non-teacher-centered) classroom has been shown to enhance student motivation and achievement. Ali, Akhter, Shahzad, Sultana, and Ramzan (2011) noted that a problem-based learning (PBL) experience in mathematics, because of its relevance and “real-world” design and approach, creates a sense of ownership of the content.
PBL requires that students arrive at a novel resolution to an ill-structured, real-life problem. In this study, however, we asked students to create rather than simply solve problems. Although the problem-creation exercise is not per se a PBL exercise, we believe that the outcomes of problem creation will be similar to those of PBL: enhanced motivation, retention, and conceptual understanding (Torp & Sage, 2002). Based on informal assessments from and conversations with instructors using variations of this exercise, students have found the sequence of assignments that we will explore in this study valuable and have reported that they were able to think more conceptually as a result of doing them, and we intend to explore students’ attitudes more deeply.
Brunkalla (2009) has asserted a relationship between the development of creativity in mathematics and both conceptual understanding and motivation. Shriki (2010) suggested that allowing for original approaches to problem solving enhances both creativity and conceptual understanding.
The present study, however, is concerned with students who are gifted or high-achieving in mathematics. According to Sriraman (2005), the study of creativity in mathematics is a very small subset of research in the field of gifted education research. Although this study does not isolate creativity as a variable, we are interested in determining whether creating mathematics problems enhances gifted students’ motivation and whether we see evidence of creativity in the type of problems they create.
Method
Participants and Setting
Participants in this study comprised primarily sophomores and juniors enrolled in a 3-year (sophomore, junior, and senior), statewide residential high school for students identified as talented in mathematics and science. Enrollment in the school is highly selective and uses an application process akin to college admissions, with SAT scores, letters of recommendation, and evidence of achievement and motivation in mathematics and science constituting the applicant portfolio. Approximately 200 students are admitted annually. Because of the specialized STEM (science, technology, engineering, and mathematics) curriculum, SAT mathematics (SAT-M) scores are a primary consideration. Typical incoming SAT-M scores of admitted students yield a mean of approximately 650. The school profile indicates that the most recent graduating class had an SAT-M mean of 713, two standard deviations above the national mean. Because of the exploratory nature of this study, we did not use a comparison group.
Participants in the study were enrolled in Mathematical Investigations (MI). MI courses are required courses for students before moving on to either AB or BC Calculus course sequence. Table 1 presents the breakdown of participants by grade and gender.
Participants by Gender, Grade.
One week prior to the administration of the instrument, we met with students in each designated class. Students were informed of the nature of the research project and the ways in which their responses would be used.
Problem Creation
Unlike most mathematics courses in which students are asked to complete a problem or set of problems to demonstrate mastery of mathematical concepts, the classes we investigated used an assessment in which students were asked to generate original mathematics problems in their area of interest. The creation of original problems is utilized in conjunction with typical homework assignments.
The problem-creation assignments consist of the writing of three original conceptual problems over the course of the semester. With the prompt that the problems be conceptual in nature (as opposed to routine problems that can be solved simply by applying a known solution method), students were encouraged to think in novel ways. For the first conceptual problem, there were some examples from a previous exam to give students an idea of what constitutes a “conceptual” problem. For the problem-creation assignment, we were interested in understanding the students’ motivation in the development of the problem, so we did not instruct them to construct problems that they thought would be engaging or motivating to others. Instead, we asked them the source of their motivation to create their original problems.
The assignment itself consisted of four parts. The following steps were taken verbatim from the assignment sheet, and the full set of instructions for the assignment can be found in Appendix A.
Motivation: How did you come up with the problem? Was it based on a problem on the worksheets? An exam? A Problem Set? Were you doodling? Did it come to you in a dream? In the shower? Just a sentence or two will suffice here. But, importantly: acknowledge your source! It is OK to look at other problems, just cite them if you use them.
Problem Statement: Fairly self-explanatory. But a caution: give it to someone else to proofread! One of the most common traps to fall into is to write a problem which can be interpreted in more than one way. Is your problem stated absolutely clearly, so that someone else can understand it perfectly without needing to ask you any questions about interpretation?
Problem Solution: Again, self-explanatory. But your solution should be in paragraph form, using complete sentences! And if you only have a partial solution, you should explain where you are stuck and those questions whose answers could enable you to make further progress.
Reflection: Only a few sentences are necessary here. What did you learn? What did you observe about yourself as a problem writer? At the end of the semester, you will need to write an essay about your growth as a mathematician and problem-writer, so making notes along the way would be a good idea.
After three of these assignments were completed, students were asked to write a brief reflective paper given the following prompts: (a) How did you grow as a problem-writer this semester? (b) Was this type of assignment valuable? Why or why not? This reflection was then submitted as part of a mini-portfolio, in which students compiled all their graded problems and their reflection in one PDF document.
Data Gathering
Instrument
We surveyed students in class immediately following the submission of two of the three problem-creation assignments. For this exploratory study, we administered a paper-and-pencil survey developed specifically to assess students’ engagement and motivation for the problem-creation assignment. The survey presented Likert-type items that were derived from several sources: (a) trends in comments from students in prior classes in which problem creation was required of students, (b) course outcomes, and (c) formative questions posed by the instructor following several years of similar class exercises.
The survey instrument consisted of 11 forced-choice and 3 open-ended questions. The open-ended responses provided an opportunity to gather information that is illustrated by supporting examples. The open-ended questions permitted students to clarify what was of interest to them and allowed us to discern emerging issues and patterns even when there was no clear evidence requiring a research hypothesis or narrowly devised set of questions. The forced-choice questions of the interview protocol, however, provided a vehicle to respond to a priori concerns or issues. The complete survey instrument is presented in Appendix B. The following sample items represent the type of open-ended questions posed to students:
Creating original mathematics problems helps me understand mathematics concepts more effectively than solving assigned problems.
I am more engaged and interested in mathematics when I am allowed to create my own problems.
Creating original problems causes me to think about my own thinking (metacognition) more.
I find creating problems more challenging than answering problems posed by the instructor.
The forced-choice questions were posed with a 5-point scale in which 1 = strongly disagree and 5 = strongly agree. The forced-choice items were followed by three open-ended follow-up questions:
Has the exercise of creating original mathematics problems enhanced your motivation in math class? If so, in what ways?
Has the exercise of creating original mathematics problems enhanced your ability to think creatively? If so, in what ways?
Have you been able to transfer any of the skills you have developed in the creation of original problems to other courses, including courses outside of mathematics? Give specific examples, if possible.
Results
Analysis of Likert-Type-Scale Data
Because we used a 5-point Likert-type scale for the survey, between-groups analyses were conducted using the Mann–Whitney U test for significance. In the analysis by class, because there were only two seniors (female), we eliminated them and did not run analyses including them.
In the first analysis, we asked whether there were gender-based differences on the individual items on the problem-creation survey. We also asked whether there were differences between classes (juniors and seniors) on the same individual items. Results of the Mann–Whitney U test did not prove to be statistically significant in between-group differences on any of the 11 questions.
The fact that we did not find any gender-based differences may, in fact, be a notable finding. In the school in which the study was conducted, a significant amount of institutional research has been conducted in the area of gender attitudes toward mathematics and science. Indeed, a significant body of literature exists examining concepts such as self-efficacy, expectancy value, and course-taking patterns in higher level mathematics courses. Our finding of no gender-based differences suggests that we examine this question more deeply in future iterations of the problem-creation exercise.
Although we treated the forced-choice items as ordinal data for the purposes of between-groups analyses, we also examined the responses using descriptive statistics and calculated means to examine the relative rank of students’ responses to individual items (see Table 2). Interestingly, the item with a significantly higher mean than all other responses (M = 3.95) was the agreement with the statement, “I find creating problems more challenging than answering problems posed by the instructor.” We found strong and varied support for this level of agreement in the open-ended responses. When we examined the ranking of the statements by mean, we found it interesting that three of the four highest ranked statements reflected challenge, enjoyment, and metacognition. Taken together, these ideas are consistent with the essential elements of optimal experience or flow, as defined by Csikszentmihalyi (1991).
Means of Responses to Problem-Creation Survey.
We were also particularly pleased with the students’ agreement with the statements that problem creation enhanced metacognition and that they seemed to derive their problems from concepts that they enjoy. Interestingly, students indicated that they generally did not create problems based on concepts with which they were having difficulty, the item that ranks lowest of all the forced-choice items (M = 2.71).
Analysis of Open-Ended Items
We were interested whether students’ autonomy in creating their own problems enhanced intrinsic motivation. Lepper and Hodell (1989) suggested that instructors can enhance autonomy by
presenting a challenging activity, which enhances self-efficacy and feelings of competence;
engaging curiosity about problems that seem complex or incongruous; and
allowing students a sense of control and ownership over their own learning.
In this section, we will analyze students’ responses to the survey’s open-ended items and attempt to identify ways in which their comments provide evidence of challenge, competence, curiosity, creativity, and control and ownership.
First, where do the problems originate? Prior to articulating problem and solution, each student is expected to describe briefly the source of motivation for the problem they created. In analyzing students’ responses to the question, we found evidence of the need for challenge, interdisciplinary thinking, and transfer to real life and feelings of competence. Many of the motivation statements suggested that current or prior work in mathematics class was significant (e.g., “I came up with this problem by analyzing what we had done so far in math class and what more could be expanded upon.”). Others suggested that the questions originated with an idea or experience in a non-mathematics class (e.g., “My motivation for the problem came from SI physics. I just finished a unit on gravitation and found it particularly interesting.”). More than 60% of responses fell into these two categories.
Other responses, however, demonstrated the variety of sources from which students find ways to explore mathematics in novel, engaging ways. A number of students indicated that the motivation for their original problems was derived from interests or activities outside of school. One student found motivation in childhood interests:
I got my inspiration for this problem by remembering the times when I was younger and fascinated by airplanes and how their various wing shapes all gave them different flying traits. Obviously triangle shape was tied into that, and I thusly decided to write a problem on triangles so that my pent up nostalgia could be freely spent.
Another student found motivation for math akin to a personal pursuit:
What made this original problem fun was the fact that I couldn’t settle on what I wanted to make my problem about. So I thought about it for a while and realized that doing math reminds me of the joy I feel when dancing. With this I sought out a way to express my dancing excellence through math and unite these seemingly incompatible concepts.
One student created a problem that originated in school:
I was thinking about the flu and everything I’ve heard about how if one person gets sick, everyone in the school gets sick, so I decided to see how long it would take for that to actually happen [in my school].
Finally, another created a problem with an entrepreneurial bent:
I came up with this problem after reading a book on marketing strategies. While reading this book, I found out that phone users typically use their phones during a specific period of day. I also found that users who mostly talk with people from another country typically receive texts in intervals [similar to people with whom they talk by phone].
No recognition of enhancement of motivation
The follow-up surveys were equally revealing about students’ motivation. Although the pre-experience survey helped us investigate the source of their problem, the follow-up questions enabled us the opportunity to investigate whether the problem-creation experience had any influence on their motivation in math class. Many of the students indicated that the problem-creation exercise had no discernible impact on the motivation in mathematics class. Their reasons for stating so, however, varied: “The problems have given an overwhelming amount of stress.” “I am motivated in math without problem creation.” “They do not necessarily relate to what we are learning in class.”
Our first response to such responses was to suggest that students found no motivation in the exercise. Each of these reasons was cited multiple times to the question regarding the effect of the assignment on motivation. Because the survey question was open-ended and because it did not provide any prompts, responses such as these suggest that a focus group protocol or an interview session might, in fact, allow students to identify ways that the exercise might be valuable (e.g., the student who created the problem integrating mathematics and dance).
It is also probable that students interpreted the term motivation in multiple ways. We approached the study with our own operationalized constructs, and we discerned evidence of such constructs. The above comments, however, might suggest that students may not recognize that motivation can relate to feelings of stress, for example. Thus, to ask students to self-assess their motivation led us to consider new or reworded prompts that were derived from responses in this exploratory study.
Evidence of challenge enhancing motivation
Specifically with respect to Lepper and Hodell’s (1989) conditions for enhancing intrinsic motivation (presenting a challenge, engaging curiosity about problems that seem complex or incongruous, and allowing a sense of control over learning), we found ample evidence of student engagement leading to an enhanced sense of efficacy, deeper engagement, feelings of ownership, and intrinsic motivation.
In several responses, we found that the challenge of problem creation led to enhanced intrinsic motivation: “It has caused me to explore intriguing math problems more intensely.” “It has gotten me to work harder.” “I tend to be more focused and give more attention to a problem I am interested in learning.”
We also found that students used the exercise to challenge themselves beyond the concepts presented in the course. The following statements, taken from different students’ responses to this question, suggest the value of self-challenge: “I explore and self-study concepts that I do not know.” “I have the need to understand the basics as quick as possible so that I can go on to more complex things.” “I begin with an original problem related to what we learned and then go above and beyond.” “Creating original math problems has made me think beyond what is in the classroom and then makes me feel better about myself.”
We also recognized in several responses that students had indeed used the opportunity to take ownership for their learning, and, in one case, shift from performance orientation (grades): “It is much more interesting figuring stuff out on your own instead of being guided.” “I am able to explore topics that interest me so I can care more about the solution. I have more motivation to solve the problem rather than just getting a good grade."
Evidence of curiosity enhancing motivation
Throughout this article, we have suggested that motivation is a process that begins with initial engagement. Thus framed, we found several instances of students who found the problem-creation exercise an opportunity to engage with math in a new and novel way: “I pick fascinating and interesting topics that allow me to get motivated to solve that problem if it is the last thing I do.” “When I come across a problem I have difficulty with, I think about similar problems I could make using that concept.”
Other findings
Mann (2006) argued, “Teaching mathematics without providing for creativity denies all students, especially gifted and talented students, the opportunity to appreciate the beauty of mathematics and fails to provide the gifted student an opportunity to fully develop his or her talents” (p. 236). In many ways, what Mann and others (see Balka, 1974; Krutetskii, 1976) asserted aligns with one of the main purposes of our study—that originality is inextricably linked to creativity. Even further, Mann and others have suggested that the construct of creativity comprises the three constructs of flexibility, fluency, and originality. Although we were not exploring fluency and flexibility in this study, we did see evidence of both and would suggest further research on this model.
One of the clear trends we discerned from the open-ended questions was the degree to which students indicated that the assignment enhanced their metacognition. The forced-choice items included a question regarding metacognition, and, although the open-ended items did not pose a similar question, we were able to discern evidence of changes in metacognition in our analysis of the question about how the exercise enhanced creativity: “It has made me more aware of my problem solving skills.” “It has helped with my questioning skills.” “Some of my original problems were related to chemistry and physics, so I could apply it to other fields of study.” “I think it has taught me to think differently and more open-mindedly about the place of math in everyday life.”
We also asked students whether or not the assignment enhanced their ability to transfer the concepts and skills to another setting. Although it is difficult to fully assess students’ actual ability to transfer the skills through a self-report survey, we were intrigued by the ways in which the skills developed through problem creation were manifested in students’ responses: “I used my original problem to find out population logistics of snowy owl populations.” “In classes that have a lot of discussion, it is now easier for me to think of my own original solutions.” “I find myself creating problems using things from everyday life. My last problem mixed Pokemon, physics, and calculus in one problem.”
Discussion
If we think of motivation as a process that begins with initial engagement and moves toward self-directed, self-regulated learning, then the value of this study has implications beyond mathematics and, indeed, beyond gifted and talented learners. Students enrolled in the school under investigation in this study are admitted on the basis of demonstrated talent and interest in mathematics and science. Furthermore, all students are required to be enrolled in mathematics. Thus, with this specialized population in mind, this study pointed us to several clear trends that support the research literature on motivation and inform classroom practice.
First, when given the freedom to design their own learning experiences, students will make connections to events, experiences, and memories of deep significance to them. We saw evidence here not only of motivational constructs such as engagement, self-regulation, and autonomy, but also of optimal experience or flow as characterized by Csikszentmihalyi (1991), such as with the student who found “that doing math reminds me of the joy I feel when dancing.” Second, although we were exploring motivation and creativity in math, students were able to suggest how the problem-creation exercise enhanced the development of their metacognitive strategies, particularly through their reflections on the creative process.
Areas for Future Research
In this study, we examined the relationship between mathematics, creativity, and motivation. Given both the design of this study and the findings posed in this article, we believe that there are rich possibilities for inquiry in a variety of areas.
First, our study focused on a novel approach to teaching mathematics. Delimiting the study to mathematics poses a number of research possibilities across domains, grades, and skill levels. What would be the student response to problem creation in disciplines other than mathematics? There are obvious parallels to the sciences (physics, which most clearly links with mathematics, and also engineering.) Similarly, how might middle school students receive such an approach? Also, would the pedagogical approach be as effective with students who are not identified as gifted in mathematics?
Second, in this study, we used self-report measures developed from several sources. We recognize this as a legitimate but limited way to assess creativity, motivation, and metacognition for our study. Such a limitation, however, leads to several research possibilities: (a) would the use of the problem-creation activity enhance these traits as measured by standardized instruments? and (b) are the effects of enhancing these traits evident in subsequent math classes?
Third, we did not include measures of student achievement as a variable. The students’ responses provided evidence of enhanced awareness of metacognitive skills, but would the students demonstrate improvement on indicators of math achievement (e.g., math grades, in-class test performance, or Advanced Placement (AP) performance)? As we reviewed the findings of this exploratory study, we realized that the promising results demanded further inquiry, and we are currently exploring students’ experiences with problem creation as it relates to course-taking behaviors, success in mathematics, conceptual understanding, and other indicators of mathematics achievement. These questions are also related to issues related to changes in mathematics efficacy, which may be influenced by performance on problem creation.
Our nonstatistically significant finding from the forced-choice items poses areas for deeper inquiry into gender and novel approaches to teaching and learning in mathematics. For example, does such an exercise mitigate differences in the value that students ascribe to mathematics?
It is important to note that, based on postgraduation surveys, we know graduates of this school tend to pursue STEM majors in college at a rate significantly higher than national norms. Subsequent study, however, might be enhanced by assessing whether those students who intend either to pursue a mathematics-related major or career or who hold deep motivation in mathematics find the problem-creation exercise most engaging and rewarding.
Implications for the Classroom
The self-directed nature of problem selection is critical to the success of the assignments. Some students write and solve problems at a level far beyond that of the course material. It would be impossible to present such students a similar challenge by giving the entire class the same homework assignment. The instructor reports that he occasionally gains new insights by reading problem solutions, as some students tackle traditional topics with fresh perspectives.
How might a mathematics teacher use such an assignment in his or her classroom? Tailoring the assignment to the teacher’s background is important. Having taught at the university level for 14 years, the instructor in this course is comfortable with a more open-ended approach, but for less-experienced teachers, the idea of having students write “conceptual” problems may be daunting. A narrower focus could still prove beneficial. For example, “Write a word problem whose solution involves the Pythagorean Theorem,” or “Create a problem involving rate, time, and distance that involves Sam riding a unicycle at a rate of 3 m/s.” In such cases, the motivation section of the assignment may not be necessary.
Assigning grades is highly individual in nature. For this assignment, the instructor used a flexible letter-grade rubric where an A meant the problem is conceptual and well-written, a B meant the problem is too routine or too many errors are present in the solution, and a C meant a lack of effort or a clear last-minute attempt at doing the assignment. Using “+” and “−” allowed for further distinction. Thus, if a student genuinely tried, he or she would not earn a low grade, but also would not earn an A unless the work was well done. Because of the wide range of assignments students turn in, a detailed rubric is not particularly helpful. Moreover, instructors need some flexibility. The instructor gave the example of a precalculus student who tackled writing a calculus problem: Although the solution had quite a few errors, the student still earned an A− as a result of a genuine and relatively successful effort to go far beyond the course material.
Teachers must also decide upon the level of detail for writing comments on problems. For more capable students, comments on details of mathematical style are appropriate, whereas for students who have difficulty solving the problem they pose, more general comments are given.
Finally, as with many types of instruction with which teachers do not have adequate experience, problem creation requires planning and a level of comfort in assessing conceptual understanding. Given the responses that the students generated and the sources of their motivation, we were both surprised and very pleased with the thoughtfulness and creativity demonstrated in their original work. And, as you demonstrate your enthusiasm for the assignment, you may generate similar excitement for creating problems among your students.
Limitations of the Present Study
This study originated with our interests in mathematics pedagogy and student motivation and creativity. We found rich, thoughtful responses from the students, and though we were pleased with the results of this exploratory study, we acknowledge several limitations.
What we did not ask in this study was how students valued and were motivated in mathematics relative to other subjects. Therefore, we do not have a measure of relative value of mathematics for each student, which may have influenced their responses on the survey.
Methodologically, this study had several inherent weaknesses. First, although this was an exploratory study, the participants were all enrolled in classes taught by the same instructor, and we could not discern whether the effects we found could be attributed to the teacher’s style or personality. The study also would have been strengthened by instrumentation that would allow comparison with normative data.
A related limitation that we acknowledge is the lack of a comparison group. Given the school’s STEM focus and the general interest and motivation in mathematics among the students, this limitation might be addressed in several ways in future studies: (a) classes might be divided in such a way that half of the students in the class are expected to create problems and the other half are not, and this would address the effect of the instructor; or (b) because several sections of the same course are offered, half of the sections might be required to pose problems, which would enhance the generalizability of our exploratory study.
Finally and perhaps most significantly with respect to the current study and subsequent use of this exercise, when we asked students about their motivation for the problems they created, our prompt (“Motivation: How did you come up with the problem?”) was open-ended and did not frame the concept of motivation in a specific way. We realize that students may have interpreted the term motivation in a variety of ways, and therefore, when we suggest that there was no recognition of enhancement of motivation, it may have been a result of a discrepancy between our own understanding of the concept and students’ understanding of the term.
Concluding Thoughts
We have shared with our colleagues, in informal ways, our design and initial findings, and we have been pleased with the interest in our study and, more importantly, their willingness to try similar exercises with their students. This exercise allows both students and teachers to take a novel approach to teaching and learning. Most importantly, we believe, it creates opportunities for teachers and students to ask questions—personal, perplexing, and meaningful questions.
Footnotes
Appendix A
Appendix B
Declaration of Conflicting Interests
The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
The authors received no financial support for the research, authorship, and/or publication of this article.