Exploring Geometric Components of CirclesChris Roberts
University of Maine at Farmington1/6/2016
EXPLORING GEOMETRIC COMPONENTS OF CIRCLES Roberts 2
As students progress through schooling systems and higher education, they
spend time analyzing, questioning and experimenting with geometric concepts both
inside and outside of the classroom. With the exploration of differing characteristics
and dimensions of shape, there is a large amount of material that can be processed
when researching these mathematical topics such as Euclidean and Spherical
geometry. Prior to exploring more complex and highly developed areas of geometry,
students in the United States must pass through years of assessment and
schoolwork that revolve around the Common Core State Standards found in
Kindergarten through twelfth grade classrooms. When researching this topic of set
standards for assessment, the United States was found to be one of the only
countries that have established levels of mathematical understanding that is met
and shared within an entire country. This aspect of individuality that revolves
around the United States mathematical curriculum is a component of Kindergarten
through twelfth grade education that is taught to both established and soon to be
educators around the country. Being established in 2009, the standards are still
fairly new and have begun to become incorporated into the new copies of
mathematical textbooks for schools such as Everyday Mathematics.
Due to the wide range of ages that are focused on in these Common Core
State Standards, this research assignment has been constructed in a manner that
splits the assessment narrative into two different concept maps to illustrate the
components that focus on circles. The first concept map, illustrated in Figure 1,
capture aspects of Kindergarten through sixth grade education. The mapping of
EXPLORING GEOMETRIC COMPONENTS OF CIRCLES Roberts 3
these standards focuses on the use of child-familiar language and all of the phrasing
was taken exactly from the Common Core narrative. Understanding the zone of
Figure 1: This concept map analyzes the components of the Common Core that connect to circles. This illustrates aspects of the standards created for students
between the grades of Kindergarten and sixth grade.
proximal understanding for the younger students in this grouping, it makes since to
have a presence of characteristics such as geometric vocabulary and the
understanding of the names of geometric illustrations. While the components of
mathematics that is present in this map can not be questioned, the aspects that are
absent can be analyzed by professionals and researchers. As the basic material of
geometry are illustrated as being an impact of early introduction to circles, there is
minimal narrations of what can come out of understanding of this topic. For these
ages, there is a strong focus on comparing the shape as an entity to other geometric
illustrations. There is no mention of gaining deeper understanding of this shape
EXPLORING GEOMETRIC COMPONENTS OF CIRCLES Roberts 4
besides the idea of using portions of the shape to illustrate fractions. This is much
different then the mentions of triangles and squares, which present a much deeper
analysis than circles in the Common Core for these ages.
The second grouping of ages to be analyzed focuses on the Common Core State
Standards for students between seventh and twelfth grade. The narratives set for
this age group can be found in Figure 2, and again this figure uses wording and
phrasing taken from the Common Core State Standards. The crucial addition found
Figure 2: This concept map analyzes the components of the Common Core that connect to circles. This illustrates aspects of the standards created for students
between the grades of seventh and twelfth grade.
within the narrative for this group is the mention of pi. With the addition of this
crucial component of circles, students are able to find calculations such as
circumference and area. The interesting part about this new concept is that the
standards to not elaborate on characteristics of area, circumference, or pi any
deeper than the idea of finding calculations by the use of formulas. In fact, formula
use is one of the only aspects of circle geometry mentioned for the students of
EXPLORING GEOMETRIC COMPONENTS OF CIRCLES Roberts 5
grades seventh and eighth. Arc lengths, radians, and theorems that relate to circles
in geometry are first mentioned for students between the grades of ninth and
twelfth. These are illustrated in bold near the bottom of Figure 2. The theorems that
are stated in the Common Core State Standards are also elaborated on in Figure 3.
Figure 3: These are the theorems and standards that are narrated in the high school Common Core State Standards for the geometry of circles.
As this area of geometry was explored, research was done to find what method of
mathematical instruction is being used in other countries around the world. One of
the countries that presented a very different method of mathematical education and
instruction was Australia. As similar components of shape and mathematics were
being taught, there was a much more abstract manner of presentation. While
looking through teaching material published by this country, a lesson plan was
found and compared to the construction of a teaching plan in the United States.
There was no mention or connection to standards that were established by the
country for the age group being taught by the classroom instructor. Relating the
EXPLORING GEOMETRIC COMPONENTS OF CIRCLES Roberts 6
plan back to the concept map for early education students, it is found that all of the
components are present. The focus of analysis with this document was the
additions found that differed from the focuses of United States mathematical
education. As mathematical vocabulary was used, such as diameter, circumference
and radius, the classroom investigation focused on how these characteristics
connected to pi. The reason for how this is different than the narrative of the
Common Core is that pi was explored and tested to deepen the students
understanding of what the irrational number stands for and represents. The
narrative of the Common Core found in the United States mentions all of this
vocabulary but the vocabulary is introduced as components of a geometric formula
for calculation.
With the exploration and research of how geometric instruction is conducted
in the United States and other countries such as Australia, a concept map can be
constructed that captures all of the components that both go into and come out of
the understanding and teaching of circles in geometry. As shown in Figure 4, there
are a lot of components that go into the understanding of circles that were
mentioned prior in the Common Core State Standards concept maps or the
Australian method of instruction. Additions are the components of deeper
understanding and knowledge of topics such as the base-ten system found in
mathematics. One other area of geometry that has been put into this concept math is
trigonometry. This is a connection to the component of geometric language and
captures an area of material and calculations that can be used to aid students
understand the ideas and functions of area and other aspects of circles.
EXPLORING GEOMETRIC COMPONENTS OF CIRCLES Roberts 7
Figure 4: This concept map is a thorough illustration of all aspects of circles in geometry found through research and intellectual conversations. Only the
components that go into the geometry of circles are narrated in this illustration.
Although the first portion of this concept map is not deeply expanded or more
elaborate than those of the Common Core State Standards mentioned earlier, the
aspects and intellectual ideas that result from the understanding of circles is much
more elaborate. With exploration of research and the communication of educated
components of both Euclidean and spherical geometry, a large amount of material is
found to come from characteristics and knowledge of circles in geometry. This
EXPLORING GEOMETRIC COMPONENTS OF CIRCLES Roberts 8
material is found in the concept map illustration found in Figure 5 below. Many
areas mentioned earlier are present in the outcome illustration of geometric
knowledge. Similar components are the use of formulas, recognition of geometric
aspects such as dilations and reflections, and the outcomes of using formulas.
Figure 5: This concept map is a thorough illustration of all aspects of circles in geometry found through research and intellectual conversations. Only the
components that result, or come out from, the understanding and familiarity of the geometry of circles are narrated in this illustration.
EXPLORING GEOMETRIC COMPONENTS OF CIRCLES Roberts 9
The two major areas of mathematics placed into this map is the components of
polygon inscription and the differing number of dimensions in geometry that lead to
both Euclidean and spherical geometry. In the time taken to research this topic,
these were two aspects not mentioned in classroom instruction. Polygon inscription
incorporates many factors mentioned throughout this paper, such as interior angles,
radians, and components of shape, but the idea begins to break down the circle and
not see the geometric illustration as a single entity. This concept of inscription
challenges ones knowledge of the characteristics and factors that make up circles.
One of the most intellectually challenging, and fascinating, area of geometry
explored with circles is the idea of three-dimensional geometry. As students become
introduced and educated about geometric concepts and ideas, it is common that
they are taught to think in a manner of a two-dimensional plain. With the abstract
idea challenging characteristics of Euclidean geometry with complex ideas of
spherical geometry, one can be more intellectually exposed to differing possibilities
of shape. These abstract areas of geometry need to be expanded on in classroom
instruction and explored to keep away from students presenting a lack of abstract
understanding and knowledge on the deep understanding of concepts of geometric
mathematics.
EXPLORING GEOMETRIC COMPONENTS OF CIRCLES Roberts 10
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