CHAPTER V
CONCLUSION AND IMPLICATIONS
Conclusion
Study Questions
The study was guided by the following questions: As an instructional starting
point, what is the role of a contextual realistic problem involving a Ferris wheel in aiding
students’ understanding of trigonometric functions? In what ways does the context of the
Ferris wheel support students’ developing understanding throughout a trigonometry
curriculum?
Summary of Methods
In order to address the above questions, a unit of trigonometry instruction was
designed around the central component of an initial project involving the animation of a
Ferris wheel through the use of the software The Geometer’s Sketchpad. Since the
researcher was also the classroom teacher, the study is best characterized as action
research. Students in three sections of high school precalculus began the study of
trigonometry attempting to define the motion of the Ferris wheel—they were not
informed of the topic of study. From this initial activity the concepts of trigonometry
naturally followed and were in fact developed by the students under the guidance of the
teacher. The motivation for this design followed from using Realistic Mathematics
Education, or RME, as the theoretical framework guiding the study.
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A fourth precalculus class, taught by the same instructor, followed the design
presented in the course textbook (Larson, 2001). In agreement with a traditional design
of placing the applications at the end of the instructional unit, the comparison class
completed the same activity at the end of the trigonometry instruction. Thus the
trigonometry unit for the study classes was the result of a reversal of the typical
instructional design.
Students in both groups completed written reflections concerning the connections
they made with the Ferris wheel and their opinions concerning how the Ferris wheel
helped, or hindered, their understanding of the unit. In addition, interviews were
conducted with 5 students at two points during the unit. The students were chosen based
upon their written reflections as well as to represent diverse ability levels—as previously
shown throughout the school year. In addition, a single student from the comparison
class was interviewed. He was chosen for his average ability level as well as his unique
comments during the written reflections.
From these sources, several themes emerged from the student comments.

Real World—a theme concerning students’ impressions and beliefs about
mathematics and especially trigonometry in regard to applicability in everyday life.

Visualization—a theme concerning the appeal of the visual nature of the problem.

Reinvention— a theme concerning the creation of significant mathematics through
the guided process.
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
Referential basis for instruction— a theme concerning the use of the Ferris wheel as a
means to anchor their trigonometric conceptions of the unit circle and graphing of
trigonometric functions.
Instructional Benefits
One primary instructional benefit followed from the element of reality inherent in
the initial problem, and thus the entire unit. The real aspect provided motivated the
study. The ability to connect mathematics to the real lives and experiences of students
afforded them great opportunities for the learning of the content.
A second primary benefit followed from the reinvention process. Students felt
ownership of the content. There was a sense of pride and autonomy concerning their
creations and the freedoms provided during the initial project. The problem scenario and
associated “discovery” activities generated experiences the students could refer back to
throughout the unit.
The Ferris wheel served initially as a model of circular motion and trigonometry,
yet as the students developed, the Ferris wheel became a model for these elements for
many students. Students used the Ferris wheel as a way to reason the graphical behavior
of the trigonometric functions. The Ferris wheel supported the concepts of amplitude,
period, and the general behavior of the graph increasing or decreasing through various
angles. Analyses of the data revealed that the students did more than simply recognized
these connections while looking at the Ferris wheel. Rather, students used the experience
of the Ferris wheel as a model for graphing and for the unit circle in a manner that they
would refer back to the Ferris wheel to make sense of future problems involving the
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graphing of trigonometric functions. Without this experience, students would be left to
whatever memory device they could devise. Such devices may be unreliable as seen in
the case of the comparison student who relied upon faulty memory devices. This was a
stark contrast to most of the other students interviewed who were from the experimental
classes.
As it concerns instructional benefits, animation played a significant role in the
influence of student understanding. The animations linked the concepts from reality to
the world of mathematics by tracing out trigonometric waves as the Ferris wheel spun.
The ability to see the motion led to more flexible considerations of concepts like the unit
circle. Students demonstrated an ability to move beyond the typically labeled values.
This led to a much more flexible implementation of the unit circle
As a whole, many students took advantage of the realistic contexts as a means to
support the study of the unit circle. In fact, the students described the Ferris wheel as a
metaphor and a building block for the study of the unit circle. Overall, many positive
connections were made by students to identities, unit circle, triangles, coterminal angles,
period, amplitude, and wave behavior to name a few. The use of the Ferris wheel and
associated animations and activities allowed for a more deeply connected conception of
trigonometry built upon experience rather than simply memory or pneumonic devices.
Evaluation of the Curriculum
As a whole, the first iteration of the curriculum created a logical reorganization of
the trigonometric concepts for precalculus classes. This curriculum incorporated several
significant differences from other precalculus sequences.
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One primary difference from other curriculums is the problem and project that
began the whole unit. Again, this design was a result of the reversal of instruction and
the tenets of RME. As already discussed, the instructional benefits are directly linked to
this introduction to the unit.
Second, adaptation and implementation of this curriculum by others is achievable.
Some elements were obviously original to this study. However, many assignments were
taken from the textbook used for the course. What distinguishes the approach is the
connection between various topics. For example, the topic of the unit circle was
motivated from right triangle trigonometry and was an obvious extension from the Ferris
wheel initial problem. Therefore, since some groups discussed a right triangle connection
for the Ferris wheel problem, it became natural to address the unit circle from this point
of view as well. Therefore, the curriculum is also distinguished by taking advantage of
the connection to reality and the experiences and constructions of the students. Others
could align the sections in various other textbooks and use a similar approach presented
here.
Finally, the curriculum is also unique for its incorporation of technological
aspects. Throughout the unit, many lessons incorporated uses of The Geometer’s
Sketchpad as a means of illustrating certain aspects of trigonometry. Students were
especially fond of the animations throughout the unit. Without the use of the animations,
demonstration sketches, or discovery-based sketches, the richness of the curriculum
would be lost. For example, the subtraction of the initial animation of the Ferris wheel
and its connection to the sine wave would undermine the entire unit. In addition, the use
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of sketches that illustrate graphical behavior and concepts within the unit circle are also
significant to the overall study.
As this was the first iteration of the curriculum, additional cycles will be
necessary to polish certain aspects. Several sketches needed to be adjusted. For
example, the initial sketch for day 4 had a construction mistake that generated errors in
the calculation of the arc length. Other sketches were adjusted aesthetically to be more
organized or pleasing. Items and information were reorganized in the viewing window or
an interactive button was added to move or hide certain elements, yet the substance of the
sketch remained intact.
Concerning the topics of study for the given days, some days took longer than the
original plan. The curriculum and corresponding information for a given day found in
appendix A represents the actual timeline of events in the classroom. This is not an
unexpected result for any type of lesson planning.
As a whole, the curriculum addressed the aspects necessary for this study. The
curriculum was built around, and driven by, the initial animation of the Ferris wheel. For
this reason, the curriculum was laid out to build up to the graphing of trigonometric
functions and the unit circle using what the students already knew about right triangles.
Implications
The results of this study suggest several important implications for the study of
trigonometry and, in general, the study of mathematics. First of all, mathematics is a
pursuit of understanding and describing real situations. The students in this study
initially believed in the lack of utility for school mathematics. As discussed, this is not a
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unique occurrence as research shows a separation between school mathematics and
reality. Teachers must attempt to find problems that will promote connections to the
world and lives of students. Without any connection to reality, students often assume that
mathematics is an indulgence into a world invented by mathematicians to torment school
children. This connection to reality must be more than simply assigning word problems
from the textbook. As illustrated by the comparison class, simply including word
problems throughout the unit does not allow for the same opportunities for connections
by the students. Therefore, educators must help students see the utility and efficacy of
the study of mathematics by using rich problems as a means for the motivation of
mathematical concepts.
In addition, the research affirms the placement at the beginning of the unit of the
real-life connection to the mathematics as directed by RME. This placement allows for
connections to be made back to the real-life problem posed to begin the unit. The end
result is the development of formulas and ideas rather than their being artificially
presented to students. Because of this, teachers also should reconsider the order of their
instruction and allow for more guided discovery and authentic problem solving to occur.
In addition, to prevent the recycling of poor instructional methods based solely upon
experience, methods instructors must continue to challenge the conceptions of preservice
teachers concerning the nature of mathematics and the design of units of instruction.
Directions for Further Research
This study suggests several additional areas requiring further study. For example,
throughout the reflections and the interviews, many comments discussed the role of
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visualization and animation. More work needs to be done to study the unique role of
animations, or motion, in the development of student understanding and how this differs
distinctly from other types of more static visualizations. For example, Can the role of the
real world and the animation be separated and be equally successful? A comparative
project could be designed to investigate three groups—animation devoid of real-life
context, animation with real-life context, and a real life context devoid of animation.
In addition, if a similar study was completed at a larger school, the issues of
interference between the classes could be eliminated. For this study, the students were
keenly aware that some of the classes were being taught in a different manner and that
only some of the classes started with the Ferris wheel problem. Such a design would
allow for more discrimination to be made between static images and the use of computer
animations.
Finally, the role of memory and the use of real-world problems building upon the
experiences of students should be investigated. Do students that learn a concept in a
manner consistent with this study maintain knowledge over time of the concepts to a
greater extent than students without such experiences?
Synopsis
In addressing the questions of the study, the evidence shows the potential power
of a contextual realistic problem-solving scenario as the instructional starting point for a
trigonometry unit of instruction. Even though not all students availed themselves to the
contextual footholds, many took advantage of the context to support their thinking
concerning the unit circle and the graphing of trigonometric functions. Since the context
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was the starting point, the Ferris wheel supported the developing understanding of the
students by allowing many students to shift from considering the animation as a model of
trigonometric functions to a model for trigonometric functions. With the rich initial
experience, students built upon this foundation to generate connected and flexible
understanding of graphing trigonometric functions and effectively using the construct of
the unit circle.