CHAPTER 2

REVIEW OF LITERATURE

    This chapter presents a review of the literature. The literature supported a qualitative study on the initial offering of the Advanced Placement (AP) Statistics course. Topics covered include theoretical construct, statistics education, AP statistics, technology in education, problem solving, and cooperative learning strategies.

Introduction

    The mathematics curriculum has remained unchanged over the last 50 years even though our society is constantly changing (Good, Murlyan, & McCaslin, 1992). Students graduating from high school today need the ability to reason and solve problems, communicate effectively by writing and speaking, and utilize technology in as many different ways as possible. Our society is more active and interactive than ever before. The demands of our current society mandates curricular change (AMATYC, 1995). To meet these demands, educators are moving from dispensing knowledge to facilitating active learning. Such a path shifts the focus on understanding rather than long, explicit derivations of meaningless mathematical results in a formal language (Cobb, 1997; Davis, 1992; Moore, 1997). Although students must develop basic mathematical skills, the processes, concepts, and understanding should take precedence (NCTM, 1989). When problem solving was integrated with active learning, deeper conceptual understandings were developed (AMATYC, 1995; NCTM, 1989). Also, AMATYC (1995) recommended that faculty provide a supportive learning environment and promote an appreciation of mathematics. In order to reflect that mathematics is an evolving, human endeavor, the way we teach and the way students learn needs to integrate interdisciplinary problem solving. According to Schoenfeld (1992) "problems can be related to real world experiences. But a more interesting aspect of problem solving involves finding patterns that can help understand the world around us" (p. 343).

Theoretical Construct

    Following Piaget’s influence, circa 1964, ideas and theories about learning changed (Ernest, 1996). Noddings (1990) described how researchers in various fields shifted from a behaviorist emphasis to a different learning theory. For example, some psychologists shifted their ontological position from behaviorism to a theory that connects cognitive activity and learning. This connection idea evolved into a learning theory called constructivism. More recently, mathematics educators and researchers applied constructivism to teaching and learning mathematics. Ernest (1996) stated that "constructivism is emerging as perhaps the major research paradigm in mathematics education" (p. 335).
    While details of this learning theory generated controversy and debate, most agreed on the basic premise that individuals constructed knowledge. Hatano (1996) suggested this construction of knowledge was unique, depending on the learners’ experiences, their ideas of reality, and their interpretation of these ideas. Therefore, each learner arrived with uniquely constructed mathematical ideas.
    Pedagogical implications were numerous and complicated. One aspect relevant to constructivist teaching was that learners constructed new knowledge by connecting new information to existing information (Cobb et al., 1990; Davis et al., 1990; Ernest, 1996; Hatano, 1996; Noddings, 1990; Shaughnessy, 1992; von Glasersfeld, 1992). If learners were expected to construct accurate ideas, teachers must consider each individuals’ preexisting cognitive structures and processes. One pedagogical goal, therefore, was to uncover students’ misconceptions and provide opportunities for them to construct (or reconstruct) sound mathematical concepts that incorporate their prior knowledge (Hiebert & Carpenter, 1992). However, creating pedagogical models that assisted students in correcting misconceptions and making these connections has been difficult. Rather than developing inflexible models reminiscent of behaviorism, constructivists have suggested pedagogical strategies that actively involve students in the learning process (Borasi, 1995; Cobb et al., 1990; Confrey, 1990; Davis et al., 1990; Ernest, 1996; Goldin & Kaput, 1996; Hatano, 1996; Hiebert & Carpenter, 1992; Noddings, 1990; Shaughnessy, 1992; von Glasersfeld, 1992).
    Adapting constructivist ideas to tangible teaching practices challenged educators. Researchers recommended a variety of ways to incorporate constructivist techniques into practical application. Goldin & Kaput (1996) discussed the importance of choosing problems and activities that represent the kinds of mathematical structures teachers want students to develop. They recommended students incorporate different types of mathematical thinking: connect mathematical meaning with respect to other domains, use appropriate skills and procedures, make abstract connections, and apply broad concepts to a smaller related set. Herscovics (1996) suggested that constructivist teachers present materials in a way that allowed students to "reconstruct, not reinvent" mathematical concepts (p. 375). He challenged teachers to remove the formal structure of mathematics that is frequently beyond students’ understanding. If a teacher could answer "What does it mean to understand" a given concept, he or she could work toward presenting the material in a way that conveys that meaning. Hatano (1996) also discussed practical implications of constructivist theory on teaching. He recommended that teachers allocate reasonable amounts of time on new information allowing students to reorganize existing structures. He argued that, even though connecting new ideas into existing thought patterns were time consuming, it was necessary if students were to construct accurate mathematical understandings. Overall, researchers agreed that teachers concerned with constructivist-based pedagogy should include activities, cooperative learning strategies, present mathematics as an interdisciplinary subject, and encourage students to use their individual skills and ideas.

A Model for Constructivist Teaching

    Confrey (1990) presented a model for constructivist-based teaching. The fundamental premise underlying the six components of her framework, was that "the skill the constructivist must truly develop is flexibility" (p. 108). She further suggested that the overall aim of a constructivist teacher is to provide opportunities for students to "develop their own cognition" (p. 110). This model was based on her belief that instruction should be designed to incorporate individual students’ views of learning. Her model required that the instructor attempt to understand the students’ background, viewpoint and misconceptions so he or she can assist the student in developing more accurate understandings.
    The first component of Confrey’s (1990) model involved students’ taking ownership of their own learning. She suggested that the instructor consistently require that students be responsible for their answers. She provided four techniques to help accomplish this task: ask students if they are right or wrong; require them to at least explain what they had attempted; act as a facilitator in problem solving; and involve students in assessing their own work. These techniques encouraged students to become aware of their thinking processes.
    As students learned to think about their own cognition, they developed reflective thought patterns. For the second element of her model, Confrey (1990) asserted that "reflection is the bootstrap for the construction of mathematical ideas" (p. 116). She presented three levels of questioning that focused on students’ awareness of their learning processes. First, the instructor asks students to rephrase or restate the problem. She explained that students cannot solve a problem until they identify what is actually being asked of them. Next, require the students to explain their work so the teacher could probe gaps in the individuals’ thinking. Guided by the teacher, students were led to reflect on their discrepancies and constructed appropriate connections. Last, after students have explained the procedure, they must defend their answer in a way that is consistent with their understanding of the problem and their procedure to produce the solution. Confrey believed this type of explanation required reflective processes.
    The third component of Confrey’s (1990) model pertained to the teacher’s ability to develop knowledge about the individual learners. To gain insight into the individual’s preconceived ideas, misconceptions, and thinking patterns, Confrey recommended that teachers work with students individually whenever possible. As teacher and student converse individually, the teacher can "try to aid the student in building a more powerful construction, from the student’s point of view" (p. 120).
    Confrey (1990) proposed "identification and negotiation of a tentative solution path" as the fourth component of her model (p. 120). She suggested, as teachers discover what was effective for one student, they attempted the same techniques with another student. However, these same techniques may not have been effective with another student and the teacher adapted to that individual’s ideas and thinking processes. She stated that the instructor must be flexible and resist rigid planning or structuring of what must occur in a given lesson. This may appear to lead to unreliable outcomes. However, the instructor can guide the discussions so that overall concepts are addressed.
    The fifth component involved reviewing the problem with the student. The purpose was to allow the student to reflect again, obtain overall understanding, and gain a sense of accomplishment. Confrey (1990) suggested this step is crucial for students to recognize future problems that might utilize similar solution processes.
    Last, the teacher must be committed to presenting the material in a rigorous manner. Using alternative ways to present and analyze concepts does not undermine what must be accomplished mathematically. Rather, alternative forms of instruction can provide opportunities for students to individually construct sound mathematical ideas.
    Overall, Confrey (1990) suggested that teachers focus on recognizing and understanding the individual’s ideas and thought processes that each student brings to the classroom. She outlined a model that provided a framework for the instructor to facilitate and direct discussion instead of lecture. The recommended techniques were designed to encourage students to take responsibility for their learning, become aware of what and how they learn, explain their course of action, and defend their answer. These techniques engaged students’ actively in the construction of their own mathematical learning.

Students’ Learning in a Constructivist Environment

    The learning theory of constructivism was student-centered. It emphasized the students’ individual view of reality, how they represented this view, and how they interpreted this view (Schoenfeld, 1992). Rather than attempting to transfer knowledge to learners, teachers focused on understanding the individuals’ ideas in order to assist them in constructing, or reconstructing, mathematical knowledge. Hatano (1996) asserted "that students construct mathematical knowledge by themselves can clearly be seen in procedural bugs and misconceptions" (p. 207). He further suggested these misconceptions evolve from their attempts to fit new ideas into existing, but conflicting, cognitive structures. Specifically, a misconception was invented by a learner as a result of trying to make sense of his or her limited experiences. Schoenfeld (1992) discussed how learning involved an overall context consisting of interlocking pieces. He stated that researchers have little understanding of how these pieces were constructed or how they were used to connect to each other. Borasi (1996) examined learning processes in constructivist environments. She concluded that students, with different abilities and backgrounds, who were taught mathematics in a constructivist manner were empowered to make mathematical connections.

Statistics Education

    As our world became more dependent upon technology and the application of rapidly changing information, statistics educators agreed that teaching statistics was increasingly more important (Cobb, 1992; Garfield, 1995; Moore, 1997; Shaughnessy, 1992). Statistics pervaded the American culture (Cobb, 1992). Citizens were bombarded with conclusions based on statistical analysis daily via the press. Corporations, all levels of government, and educational organizations made decisions based on the results of statistical studies. The increased use of statistics called for greater statistical understanding (Cobb, 1997; Moore, 1997; Shaughnessy, 1992). According the Shaughnessy (1992), whether we teach it or not, people will use and misuse statistics probably more than any other field of mathematics.
    Consistent with the current reform movement in mathematics education, statistics educators and researchers challenged teachers to examine their teaching practices. They recommended that teachers teach in a way that addresses different learning outcomes. Cobb (1992) suggested the goal in a statistics course was for students to emerge with the ability to read and understand statistics that are available in scientific publications, newspapers, and television. Shaughnessy (1992) wrote that all our students eventually become consumers and citizens in a society that uses statistics to communicate information and influence the masses. For educators, it was apparent the role and flavor of teaching statistics must reflect the challenges that society placed on our students and graduates (MAA, 1991; Velleman & Moore, 1996).
    Cobb (1997) suggested that a statistics course offers opportunities to investigate real world situations even though many courses fail to accomplish this. He further stated too many statistics courses were presented like traditional mathematics courses that stressed theory. He acknowledged that many teachers perceive that a "good" statistics course emphasized mathematical rigor rather than analytical thinking. However, researchers involved in statistics, envisioned statistics as a course in problem solving, rather than just traditional subject matter (Cobb, 1997; Garfield, 1995; Shaughnessy, 1992). Garfield (1995) described advantages of teaching from a problem solving perspective. If analysis was stressed, she claimed, then students learned that it was important to make claims, determine if their claims are appropriate, and defend their claims.
    Cobb (1997) suggested that statistical thinking was different from what was presented in traditional mathematical courses. Cobb and Moore (1997) explained that statistical analysis was always within context. Data were not just numbers, but numbers embedded within the context of a specific problem. Scheaffer (1992) expanded this and suggested that these contexts must be real to the students. He rejected the practice of investigating data from contrived textbook examples. He explained that for students to intelligently explore data, they do not need extensive knowledge of statistical methods, but rather, practice at deep analysis of legitimate, real world questions. Scheaffer’s reasoning was based on his belief that "the most important decisions in a student’s life will involve data. Thus it is initially important that students be taught to gather data intelligently and look carefully and critically at all data presented to them" ( p. 70). Cobb (1992) revealed a similar idea.