CHAPTER 1
THE PROBLEM
This chapter presents an overview for a qualitative research study
investigating the initial offering of the Advanced Placement (AP) Statistics course.
Topics include problem, research questions, rationale, theoretical construct and overview
of the study.
Introduction
Nationwide efforts in mathematics education reform were stimulated
in 1989 when the National Council for Teachers of Mathematics (NCTM) released the Curriculum
and Evaluation Standards for School Mathematics and the National Research Council
(NRC) published Everybody Counts: A Report to the Nation on the Future of Mathematics
Education. The spirit of reform was inspiring mathematics educators and researchers to
study the ways students think and learn. This reform included a shift in the kind of
mathematical knowledge students were expected to develop from formulas and memorization of
algorithms toward constructing individual, mathematically sound concepts and
understandings (Davis, 1992).
Many mathematics education researchers utilize a constructivist
epistemology to understand students’ reasoning processes and their development of
concepts. Constructivism, a theory of learning, embraces the fundamental idea that
learners construct knowledge. This learning theory also has pedagogical implications.
Rather than emphasizing traditional lecturing methods, teachers provide "a set of
mathematical experiences which the teacher attempts to arrange for the student"
(Lochhead, 1991, p. 76). This shift in pedagogy has also impacted mathematics education
research. Many education scholars across all academic levels are attempting to observe,
understand, and describe the process of learning. Many mathematics education
researchers’ goals are moving toward understanding students’ thinking about
mathematics and their construction of ideas.
The Problem
For many years, the College Board, public schools, and participating
colleges have collaborated to provide college level courses to high school mathematics
students through the Advanced Placement (AP) Program (College Entrance Examination Board,
1996). The AP program serves three educational groups: (a) students who wish to earn
college credit while in high school, (b) secondary schools who encourage their students in
the same, and © the colleges that grant credit to these students (Watkins, Roberts,
Olsen, & Scheaffer, 1997). Students enroll in AP classes their junior or senior year.
If they pass the corresponding nationally standardized AP exams in the spring, they may
receive college credit for the course. In the fall of 1996, an AP course was implemented
in Statistics. The AP Statistics curriculum provides college bound students the
opportunity to study statistics in an academically challenging high school environment.
Consistent with NCTM’s Curriculum and Evaluation Standards for School Mathematics(1989)
and the current reform movement, the AP Statistics course emphasizes concept-oriented
instruction, writing, group projects, computer applications and simulations. According to
the College Board Advanced Placement Course Description (1997),
The AP Statistics course lends itself naturally to a mode of teaching that engages
students in constructing their own knowledge . . . Important components of the course
should include the use of technology, projects and laboratories, cooperative group problem
solving, and writing as a part of concept-oriented instruction and assessment. This
approach gives students ample opportunity to think through problems, make decisions, and
share questions and conclusions with other students as well as the teacher. (p. 11)
In the same spirit of reform as that prevalent in mathematics education, the Georgia
Pre-School through Post-Secondary Education (P-16) Initiative (Vollmer, 1996) provides
opportunities for systemic reform at all levels of education. Among P-16's five purposes,
three are addressed by the AP Program. They are: (a) to improve student achievement, (b)
to help students move smoothly from one educational system to another, and (c) to ensure
that all students who enter the University System are prepared to succeed. The AP Program
is a cooperative effort between secondary schools, colleges, and the College Board
(College Entrance Examination Board, 1996). Characteristics of the AP Program (College
Entrance Examination Board, 1996) include,
A commitment to academic rigor, a reliance on outstanding secondary school teachers and
college faculty for the maintenance and development of the program, and a belief that able
secondary school students can profitably handle college-level material. (p. 8)
Clearly, the goals of P-16 are addressed by the Advanced Placement Program.
Increased attention on how students learn is changing the role of the
teacher in mathematics classrooms. Also, there is additional consideration on statistics
in the education curriculum. Meeting both these changing trends, the College Board
implemented an AP Statistics class in the fall of 1996. However, there was no research on
the effects of concept-oriented instruction on students’ learning in the AP
Statistics classroom.
Research Questions
Despite increased attention to the importance of statistics in high
school and college curricula, many educators state that little research on teaching and
learning statistics has been conducted (Cobb, 1992; Garfield & Ahlgren, 1988;
Shaughnessy, 1992). Also, due to the rapid and recent technology explosion, more research
is recommended regarding the impact of non-traditional pedagogy on learning statistics
(Dunham & Dick, 1994; Moore, 1996; Shaughnessy, 1992).
This study used ethnographic methods, including longitudinal fieldwork
and in-depth interviewing, to understand and describe students’ learning in a
concept-oriented, activity-based, AP Statistics classroom. The research focused on how the
AP Statistics Test Development Committee’s recommended pedagogy effects
students’ learning and subsequent performance on the AP exam. To achieve this goal,
the following questions were addressed. Questions 1 and 2 emerged from the study. The
emergence of these questions is described in chapter 4.
1. What are the salient features of the AP Statistics course and the genesis of these
features?
2. What teaching strategies are effective in implementing the recommended pedagogy?
3. How do projects that employ cooperative group problem solving and writing influence
students’ development of statistical concepts?
4. How does gathering data effect students’ understanding of statistics?
5. What components of activity-based, concept-oriented instruction effect
students’ performance on the AP exam?
Rationale
The reform movement within mathematics education has pedagogical
implications leading some teachers to change teaching techniques. Still, many teachers
present mathematics by lecture and assess students’ learning using traditional
methods. Teaching and assessing rote manipulations based on regurgitation of these
"taught" ideas implies that mathematical thinking is simple memorization. Many
reform oriented educators resist this notion and encourage critical reasoning about
mathematical ideas (American Mathematical Association of Two-Year Colleges [AMATYC], 1995;
Confrey, 1990; Davis, Maher, & Noddings, 1990; Mathematical Association of America
[MAA], 1991; National Council of Teachers of Mathematics [NCTM], 1989). Calculators and
computers can do the arithmetic. The student, however, needs to have conceptual
understanding in order to create or choose the appropriate technique to solve a problem
(Cobb, 1992; Creswell, 1994; Garfield & Ahlgren, 1988).
The role of statistics in education is also changing. Many educators
and statisticians see statistics as a branch of mathematics easily adaptable to the
constructivist theory of learning. As the science of data analysis, statistics
incorporates critical examination of the data, interacting with data by modeling and
hypothesis testing, and drawing conclusions based on the same. These elements of
statistical thinking and analysis involve active involvement by the student. Students need
to examine the situation, generate ideas about the behavior of the situation, and use
other statistical techniques to test their ideas.
The role of statistics is also changing in education because statistics
classes are gaining more attention and enrollment (College Entrance Examination Board,
1997). Educators are more aware that high school graduates need the ability to critically
read and analyze statistics prevalent in everyday life (Cobb, 1992; Garfield &
Ahlgren, 1988; Moore, 1990, 1997). In addition, while many students do not have a high
school statistics background, enrollment in college statistics is larger than that for
calculus (Loftsgaarden, Rung, & Watkins, 1997). Increasingly more departments,
including business, nursing, psychology, and social work, require at least one statistics
course toward degree requirements. Many students, however, who pass the course lack the
ability to apply any conceptual understanding to nontextbook scenarios. Despite the
increasing emphasis of statistics in college curricula and the growing trend in
mathematics education to develop constructivist teaching techniques, the need exists for
more research investigating the influence of constructivist pedagogy on students learning
statistics (Cobb, 1992; Garfield & Ahlgren, 1988; Shaughnessy, 1992). Specifically,
there is no research on the Test Development Committee’s recommended pedagogy’s
effect on high school students’ conceptions of AP Statistics topics.
Theoretical Construct
The learning theory of constructivism supports ontological
assumptions that "emphasize the importance of understanding the processes through
which human beings concretize their relationship to their world" (Morgan &
Smircich, 1980, p. 493). Constructivism asserts that learners construct knowledge and
understand their world by connecting new information with existing knowledge (Piaget,
1964). The mathematics education interpretation of constructivism provides a description
of the cognitive processes involved when students’ connect mathematical ideas (Cobb,
Wood, & Yackel, 1990).
Constructivism has had a profound effect on mathematics education.
Rather than emphasizing test scores as a measure of learning, many mathematics educators
are looking into students’ thought processes and how learners come to know. Seeking
to understand and interpret is consistent with the learning theory of constructivism.
Researchers emphasize identifying themes that explain how students assimilate new
information. Misconceptions and prior knowledge are important pieces of the puzzle. The
increased interest in how students construct meaning has many researchers and teachers
shifting their attention from passive to active learning. Teaching strategies are moving
from lectures of routine, meaningless computations to student activities involving
applying and understanding concepts (Confrey, 1990).
As a result of learning experiences, students’ knowledge is
constantly in flux and is influenced by the individual’s lens through which he or she
views the world (Hatano, 1996). Therefore, each learner arrives with uniquely constructed
ideas about mathematics. Students have different learning styles, various levels of
understandings, and distinct inaccurate ideas. One challenge for teachers is to uncover
misconceptions and provide activities where students can construct (or reconstruct) and
understand sound mathematical concepts. Constructivist teachers understand that a critical
element of the constructivist theory is the idea that students connect new information to
existing information (Piaget, 1964). Constructivists suggest students be actively involved
if they are expected to successfully make these connections. The challenge has been to
create pedagogical models that assist students in making these connections. Rather than
developing inflexible models reminiscent of behaviorism, constructivists suggest
pedagogical strategies that actively involve students in the learning process. Recommended
activities include using technology, projects, cooperative group problem solving and
writing assignments. The AP Statistics Test Development Committee, appointed by the
College Board, recommends these same pedagogical techniques for the AP Statistics course.
Consistent with constructivist theory, this study investigated the statistical
understandings that AP high school students construct in a concept-oriented class with
respect to the Test Development Committee’s recommended pedagogy.
Overview of the Study
Shulman (1988) argued that flexibility is required to investigate a
complex environment and school is complex. A researcher interested in identifying and
describing complex learning processes needs flexibility for patterns to emerge.
Methodology sensitive to describing complex factors encouraged adaptation of ideas and
influences (Lincoln & Guba, 1985). Qualitative researchers explored complex processes
using inductive logic (Creswell, 1994). Specifically, an ethnographic design supported the
emergence of patterns and themes. Therefore, educational researchers believed qualitative
research methods were appropriate for research that investigates complex processes
(Creswell, 1994).
To address the research questions, I contacted a teacher in a school
that offered AP Statistics. I had a limited number of schools to choose from since this
initial course was not offered at every high school. I chose a teacher who piloted the AP
Statistics class in the school year 1995-1996. The purpose of the fall quarter visits was
to observe, gather baseline data, and attempt to establish rapport. Data were collected on
students’ projects winter and spring quarters. Project notes were collected routinely
throughout their project assignment. I conducted interviews after completion of their
project presentations. The interviews sought to clarify how various elements from the Test
Development Committee’s recommended pedagogy, including the use of technology,
projects and laboratories, cooperative group problem solving, and writing as a part of
concept-oriented instruction and assessment, affected their project development.
Certain data sources included copies of students’ tests and other
work, especially project notes, that might reflect their concept development.
Specifically, all students collaborated with other students on a project. Their assignment
included submitting weekly progress reports. The students responded to a series of
open-ended questions to address. In order to check their conceptual development, these
questions required them to compose sentences and paragraphs. In addition, traditional
tests and individuals’ scores, writing samples, computer assignments, and final
project reports were collected. Data were gathered and analyzed in an attempt to clarify
how various elements from the Test Development Committee’s recommended pedagogy
affected the students’ concept development.
I conducted interviews asking questions that focused on how they did,
or did not, utilize various techniques to construct understandings. The techniques of
interest are those recommended by the Test Development Committee, including the use of
technology, projects and laboratories, cooperative group problem solving, and writing as a
part of concept-oriented instruction and assessment. Key informants were chosen for
further investigation of their learning process as patterns begin to emerge. Choosing
informants for interviewing is "largely opportunistic" (Lincoln &
Guba,
1985, p. 259). As the researcher, I observed the class dynamics seeking to identify
students who provided a variety of learning prospectives. I requested the instructor to do
the same. As I began to narrow the list of prospective informants, I discussed my ideas
with the teacher.
I utilized member checking, triangulation of data, and peer debriefing
to verify descriptions. The idea is to examine various types of collected data to identify
trends and patterns. To verify emerging patterns, the instructor reviewed drafts. Also,
these patterns were confirmed during interviews with the students. Ongoing member checking
served to verify that these patterns and behaviors were accurate reflections of the
informants’ thoughts and conceptual development.
Two research questions emerged from the data. To address the first
emergent question, I gathered a variety of resources from the College Board, the
Educational Testing Service (ETS), and the instructor. College Board and ETS resources
included published materials and structured interviews with representatives from both
organizations. To answer the second emergent question, I referred to the structured
interview and numerous other conversations with the instructor.
Summary
The College Board implemented an AP Statistics class to address the
needs of the changing curriculum of colleges. The AP Statistics Test Development
Committee, appointed by the College Board, recommended pedagogical techniques supported by
the constructivist learning theory. These techniques included the use of technology,
projects, cooperative group problem solving, and writing. This study utilized qualitative
research methods to observe, understand, and describe the impact of the Test Development
Committee’s recommended pedagogy on AP students’ development of statistical
concepts. The next chapter presents a review of the literature.
