WHAT CONSTITUTES A MATHEMATICALLY RICH AND MEANINGFUL TASK: PERSERVICE ELEMENTARY SCHOOL TEACHERS' PERCEPTIONS

Esther M.H. Billings	David B. Klanderman
Grand Valley State University	Trinity Christian College
billinge@gvsu.edu	Dave.Klanderman@trnty.edu

Abstract: This study examined the criteria that preservice teachers use to determine whether a given task is rich and mathematically meaningful. In addition, cases were used to explore the various perceptions that emerged and how they evolved over the course of a semester. Nineteen preservice elementary school teachers, all of whom were receiving either a major, minor, or emphasis in mathematics, participated in the study. Students submitted a variety of written artifacts in which they were asked to both generate and evaluate a variety of mathematical problems. Six archetypes of student responses, along with several general characteristics of student perceptions, were identified.

Significance of Problem and Theoretical Framework

Current reform movements in mathematics education emphasize the need for students to be engaged in solving rich, open-ended problems (NCTM, 2000). Since students learn from the tasks they are given in class (Doyle 1983; 1988), the nature of the task plays a critical role in determining the type of mathematical understanding that will occur in the mathematics classroom (Hiebert et al., 1997). “Rich” tasks need to extend beyond computational problems (Lampert, 1990) and should enable students to formulate an understanding of mathematical concepts that integrates both conceptual and procedural knowledge. Utilizing the work of Hiebert and Lefevre (1986, pp. 3-8), we view conceptual knowledge as knowledge rich in relationships, a connected web of knowledge. Procedural knowledge is characterized as symbolic representations (formal mathematical “language” of symbols), rules, algorithms, or procedures used to solve a mathematical task. In addition, drawing from ideas in constructivist theory (e.g., Glaserfeld, 1991; Noddings, 1990), we feel that in order for students to acquire this holistic mathematical knowledge, they must dynamically interact with and construct an understanding of a concept as they engage in different tasks.

The types of tasks that students engage in are for the most part determined by the teacher as she selects and/or creates these different problems. Consequently, if teachers do not have a view consistent with the current reform movement regarding the nature of a rich mathematical task, then it is unrealistic to expect them to carry out the intentions of mathematics education reform that emphasizes a deep conceptual and procedural understanding of mathematics. Few studies exist that specifically investigate how preservice teachers judge what constitutes a good mathematical problem. This study extends the current body of research by examining how preservice elementary school teachers judge mathematical tasks.

Research Methodology and Data Collection

The researchers used qualitative research methods, and in particular, case studies, to explore preservice elementary school teachers' perceptions of what constitutes a "rich" and mathematically meaningful task. Nineteen preservice elementary school teachers enrolled in an upper division mathematics class (investigating algebraic concepts and related pedagogical issues) at a Midwestern mid-sized university participated in the study. They were told that participation (or lack thereof) in the study would not affect their final course grade. All of the students in the class gave written permission to use their work in this study. The preservice teachers were of a traditional college-age and pursuing a major, minor, or emphasis in mathematics

Data for this study were collected in a variety of settings to enhance the validity of the research findings (Denzin, 1978). Artifacts of written work including copies of exams, homework assignments, and portions of a function portfolio submitted by students were gathered. In addition, since the first author was the instructor for the course, she was a complete participant observer and the second author was an occasional nonpassive participant observer (Spradley, 1980) of the students in the classroom. Furthermore, one class session in which students spent one hour and fifteen minutes discussing and evaluating two problems was videotaped and analyzed. Finally, the researchers discussed students’ responses to questions on exams and assignments, and made field notes on these discussions. Triangulation (Denzin, 1978) was incorporated into the data collection process.

The researchers systematically and rigorously organized, synthesized, and categorized the data using the methods of Spradley (1980). This type of data analysis enabled us to make the transition from asking general questions about the study to asking and answering more specific questions that stemmed directly from the data. Categories of criteria used to evaluate problems emerged through our systematic analysis and reanalysis of the data. Cases were then utilized to present a descriptive, holistic view (Merriam, 1985) of the preservice teachers' perceptions and how they evolved/changed over the semester. Of the 19 subjects, six were chosen as cases for this study since they provided informative and representative snapshots of the analyzed data.

Data Analysis and Results

Through our data analysis, we identified a total of six different categories of criteria used to determine if a problem was rich. These criteria were used on multiple occasions and by at least five students. The categories included multiple representational modes, patterns, variables and change, real world context, “how” and “why” questions, and connections. Within each category, a number of subcategories were also identified. Typically, when the students evaluated a problem, they focused (nearly) all of their argument on the general characteristics of the problem and spent little time evaluating the mathematics inherent in the problem.

We now provide a brief synopsis of the six archetype cases that emerged as the preservice teachers' perceptions were analyzed over the course of a semester. First, many demonstrated minimal change; submitted problems and corresponding critiques of these problems continued to be either weak or strong throughout the semester. Jill represented one fifth of the preservice teachers who had a very limited understanding of the mathematical content in a problem and consequently could not provide "good" examples of rich problems nor give a substantial argument as to why a particular problem was rich and mathematically meaningful. Her criteria focused on general characteristics of a rich problem such as "making students think" and did not examine issues related to mathematical content. Tanya represented another fifth of the preservice teachers who experienced minimal change throughout the course of the semester. Tanya, a very strong mathematics student, consistently submitted very rich and conceptually-based problems that linked together a variety of ideas. She also explained with clarity why these problems were rich and meaningful. Through the course of the semester her problems maintained a high level of quality but her explanations became more specific and fine-tuned, focusing on both mathematical aspects as well as general characteristics of the problems submitted. One third of the students demonstrated inconsistency in change. In general, they showed improvement in particular problems and/or critiques as they received written and verbal feedback from the instructor of the course. However, no consistent general patterns of change were observed.

About one fourth of the preservice teachers showed evidence of observable change. One type of change was in the language that was used to critique and analyze problems. Sam exemplifies this type of change. He was a procedurally strong student used to doing well in mathematics classes, and he consistently submitted very procedural problems. However, as the class read and discussed various articles, he began to incorporate the language and vocabulary of the articles into his problems and justifications as to why they were rich and mathematically meaningful. However, no structural change occurred in his problems. Two students demonstrated change in the mathematical and general content of submitted problems. However, the strength of the problem submitted was dependent on the source of the problem since the students were allowed to adapt problems discussed in class, found in texts, or supplementary materials. For example, Jessica M. submitted problems that were entirely procedural and problems that integrated conceptual and procedural knowledge. However, her analyses of why these problems were “rich” and mathematically meaningful were very similar. She could not differentiate between the quality of problems she was submitting. The final change observed in two students dealt with the type of critique and analysis given of a particular problem. Jessica S. and Marie began the semester by submitting problems that were quite conceptual; however, they had difficulty articulating why these problems were mathematically rich and meaningful and tended to focus on more general characteristics of the problem and did not focus on mathematical content. By the end of the semester, their problems were even stronger and both could more clearly articulate why problems were rich. However, most of Marie’s analysis continued to center on general characteristics of a problem though she became more specific in her mathematical analysis. Jessica S. focused on both general characteristics and mathematical characteristics, such as patterns.

Conclusions and Implications

We draw several conclusions from this study. First, students had difficulty evaluating problems from a mathematical viewpoint and tended to focus their analysis on general characteristics of the problem. In particular, few students mentioned or made meaningful connections between different function representations. Second, students with strong problems and critiques tended to cite criteria such as “extend and make predictions based upon patterns” and “make connections between different representations.” Finally, some students demonstrated a mismatch of words and actions (e.g., a strong critique associated with a weak problem).

We believe that this study also has several important implications for the teaching and learning of mathematics. First, mathematical understanding is necessary but not sufficient for identifying mathematically meaningful and rich problems. Similarly, knowledge of mathematical content alone does not imply an ability to create or pose rich mathematical problems for students. Second, as teacher educators, we need to be sensitive to both the mathematical tasks created by preservice teachers and their analysis of these tasks. Furthermore, teacher education programs must devote time to explicitly explore the nature, the creation, and the analysis of rich and mathematically meaningful task. This preparation is important because teachers play a critical role in providing the necessary guidance and connections between these tasks (Lampert, 1991; NCTM, 1991). Finally, research has also shown that preservice teachers’ beliefs about learning and teaching mathematics are formed during their own personal schooling and mathematical experiences (Ball, 1988). They bring these beliefs as well as their existing knowledge of mathematics with them as they enter teacher education programs. Thus, preservice teachers need to have experiences in which they are encouraged and expected to evaluate problems to determine if they are truly mathematically meaningful.

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