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The Application of Standards to the
Elementary Classroom
I made every attempt in my Geometry Unit to keep in mind
the national and state standards for elementary mathematics. In the following
discussion, I will examine the specific applications of the national geometry
standards for grades 3-5, as described by the National Council of Teachers
of Mathematics, to by geometry unit. My unit consciously tried to incorporate
standards from areas outside of geometry as well, but I will focus on
the geometry units. I use the national standards rather than the Pennsylvania
standards because I believe that the national standards encompass the
state standards and are better applicable to teaching mathematics throughout
the country. I break down the standards listed on their
webpage and describe how I used or did not use each standard in my
unit.
"Instructional programs from prekindergarten
through grade 12 should enable all students to analyze characteristics
and properties of two- and three-dimensional geometric shapes and develop
mathematical arguments about geometric relationships"
"In grades 3–5 all students should identify, compare,
and analyze attributes of two- and three-dimensional shapes and develop
vocabulary to describe the attributes. "In my unit, I put a
great emphasis on comparing different two dimensional shapes. From the
first mention of polygons in the classroom, I had students compare polygons
to non-polygons to explore the similarities and differences between
them in order to come up with a definition of "polygon."
"In grades 3–5 all students should classify two- and
three-dimensional shapes according to their properties and develop definitions
of classes of shapes such as triangles and pyramids" Most of
the lessons at the beginning of the unit were focused on building the
definitions of types of polygons, types of triangles and quadrilaterals.
The students in fact developed their own definitions which we refined
as a class for many of these shapes.
"In grades 3–5 all students should investigate, describe,
and reason about the results of subdividing, combining, and transforming
shapes." I had students do this when ever they used Tangrams
and during our lessons on transformations.
"In grades 3–5 all students should explore congruence
and similarity." I had students work with the concepts of similarity
and congruence through exploring a multitude of activities in which
they applied these principles.
"In grades 3–5 all students should make and test conjectures
about geometric properties and relationships and develop logical arguments
to justify conclusions." I had students do this on several
occasions. I often had students justify their answers to a question
such as "Can a triangle have more than one right angle?" using
a picture or diagram.
"Instructional programs from prekindergarten
through grade 12 should specify locations and describe spatial
relationships using coordinate geometry and other representational systems"
"In grades 3–5 all students should describe location
and movement using common language and geometric vocabulary."While
the unit did not focus on coordinate geometry, in discussing the different
geometrical transformations we fulfilled the "movement" part
of this objective. Students learned mathematical names for these transformations,
but also used their own language to describe them.
"In grades 3–5 all students should make and use
coordinate systems to specify locations and to describe paths."
This goal was not addressed in the unit as we did not examine coordinate
geometry at all. In ten scope of the school year, however, this would
be a goal that I would want to touch on.
"In grades 3–5 all students should find the distance
between points along horizontal and vertical lines of a coordinate system."
This goal was not addressed in the unit as we did not examine coordinate
geometry at all. In the scope of the school year, however, this would
be a goal that I would want to touch on.
"Instructional programs from prekindergarten
through grade 12 should apply transformations and use symmetry to analyze
mathematical situations"
"In grades 3–5 all students should predict and describe
the results of sliding, flipping, and turning two-dimensional shapes."
Students did this during our initial conversations about these transformations,
as well as when they made diagrams to show the effects of a turn, slide
and flip using construction paper.
"In grades 3–5 all students should describe a motion or
a series of motions that will show that two shapes are congruent."
This goal was addressed when students described how to move the different
tetrominoes to demonstrate that two tetrominoes were congruent.
"In grades 3–5 all students should identify and
describe line and rotational symmetry in two- and three-dimensional
shapes and designs." We worked a lot with line symmetry in
the classroom, including an activity in which students designed their
own symmetrical patterns using pattern blocks. We did not examine rotational
symmetry explicitly, though it was mentioned in the website students
used independently to explore line symmetry.
"Instructional programs from prekindergarten
through grade 12 should use visualization, spatial reasoning,
and geometric modeling to solve problems"
"In grades 3–5 all students should build and draw geometric
objects." Students did this on several occasions using pen
and pencil,geo-boards, tangrams and pattern blocks.
"In grades 3–5 all students should create and describe
mental images of objects, patterns, and paths." I did not address
this specifically.
"In grades 3–5 all students should identify and build
a three-dimensional object from two-dimensional representations of that
object." I did not address three-dimensional objects in an
in depth way due to time and its placement in another chapter in the
school curriculum.
"In grades 3–5 all students should identify and draw a
two-dimensional representation of a three-dimensional object."
I did not address three-dimensional objects in an in depth way due
to time and its placement in another chapter in the school curriculum.
"In grades 3–5 all students should use geometric models
to solve problems in other areas of mathematics, such as number and
measurement." Students did this when I asked them to solve
word problems in which geometrical figures would be useful.
"In grades 3–5 all students should recognize geometric
ideas and relationships and apply them to other disciplines and to problems
that arise in the classroom or in everyday life." As much as
possible, I tried to stress the applications of geometry in the student's
lives. I actually caught students using mathematical terms to describe
the type of line they made as they lined for lunch one day, which I
also feel addresses this goal.
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