Keywords: Technology, Geometry, Activities
Ref: MiriamN1
Author(s): Dwyer, Marlene C.; Pfeifer, Richard E.
Date: 1999
Title: Exploring Hyperbolic Geometry with The Geometer's Sketchpad
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 92, October, pp.632-637
Reviewer: MiriamN
Date of Review: 1/30/00
This article describes activities that can be done using The Geometer's Sketchpad to explore some principles of hyperbolic geometry. First the fundamental difference between hyperbolic and Euclidian geometry is explained. In hyperbolic geometry, all the axioms of Euclidean geometry are retained except for the parallel postulate, which is replaced by the hyperbolic postulate: "Through a given point P, not on a given line n, can be drawn more than one line that does not intersect the line n. " Brief references are made to the history of this field of geometry. One of the most popular models of hyperbolic geometry is the Poincare (accent above the "e") disk model, which is what this Sketchpad program is based on. In this model, you begin with a circle in the Euclidean plane, where all the points inside the circle are points in the model. The "lines" in the model are arcs of (Euclidean) circles that meet the boundaries of the circle at right angles, as well as straight line segments through the center of the circle. Inside the circle, most other geometric objects are defined as usual, e.g., triangles are defined as 3 hyperbolic line segments joining 3 (hyperbolically) noncollinear points, etc. After this brief explanation of some fundamental properties of the Poincare model, the remainder of the article is spent presenting several constructions and activities. It is recommended that the reader download the program from the website "forum.swarthmore.edu/sketchpad/gsp.gallery/poincare/poincare.html" in order to follow along with the described exercises. The activities involve exploring the hyperbolic parallel postulate, hyperbolic triangles and circles, exploring the circumcenter and circumcircle, and investigating properties of the centroid, orthocenter, and incenter. Several other topics are also mentioned, although details of those investigations are not provided. The authors feel that the purpose of these activities is to present a different type of geometry to students than what they are traditionally exposed to in the classroom and allowing them to compare and contrast the hyperbolic and Euclidean systems, and thus broaden and abstractualize their conception of geometry. This article aroused my curiosity, since I have never dealt much with hyperbolic geometry in any of my coursework. It is exciting to know that there is such accessible software out there to explore this area - it should be of interest to teachers and students alike. Since I have such limited experience with this subject myself, I find it difficult to critique the content of the exercises. As interesting as they seem, however, the topic strikes me as fairly abstract and advanced, and may be best suited for gifted and talented programs or even college geometry.
Keywords: Teaching Strategies
Keywords: Teaching Strategies
Keywords: Inquiry, Geometry, Research
The author believes that mathematics instruction based on the research
of the van
Hieles can help students in the middle grades succeed in high school geometry.
The van
Hieles proposed that there are five levels of mathematical thinking: 0)
Concrete, in which the
student identifies and operates on concrete geometric figures; 1) Analysis, in
which the
student analyzes properties of figures through observation; 2) Informal
deduction, in which
the student formulates generalizations and develops informal supporting
arguments;
3) Deduction, in which the student proves theorems deductively and understands
the geometric
system; and 4) Rigor, in which different postulational systems are studied and
compared. The
van Hieles argue that students within a given classroom may be at a variety of
levels in their
thinking, and that progress in learning is hindered when the level of the
student is not aligned
with the level of the material being taught. Although geometry in the middle
grades is
traditionally taught at the level of informal deduction, many students of this
age may still be in
the analytical, or even concrete stages of thinking.
Students in the middle grades often have a weak understanding of the
concepts of
perimeter and area, and consequently confuse these two terms and apply them
incorrectly in
formulas when solving problems. The author describes a van Hiele-based lesson
designed to
clarify understanding of these concepts and help students at different levels of
thinking
progress to the next higher level. Students conduct an investigation in which
they are
provided with an arrangement of square tiles and asked to add tiles to achieve a
given
perimeter. This experiment was performed by three sixth-grade students at three
different
levels of geometric thinking during a summer enrichment program. Students were
first asked
to explore the problem individually. Then through a group discussion
facilitated by the
teacher, students shared their thinking with one another, worked together to
solve related
extension problems, and gained new insights into different properties and
relationships
between area and perimeter. Through careful questioning and suggestions on the
part of the
teacher, by the end of the lesson, each student had begun to demonstrate some
characteristics
of the next level of geometric thinking. The author's belief is that use of the
van Hiele method,
both in terms of the sequence of instruction (information-gathering, guided
discovery,
explication, free orientation, integration) and knowledge of different levels of
thought, were
critical to the progress these students made on this topic.
I found little to argue with in this article. I liked the activity very
much, since it not only
solidifies concepts of perimeter and area, but is interesting and beneficial for
students at
different levels of thinking and can lead to good conjectures and further
extensions in many
different directions. I agree that knowledge and application of the van Hiele
levels of thinking
and phases of investigation can enrich instruction of geometry and other
mathematical
subjects. My only comment, and the author would probably agree, is that moving
students
from one level of thinking to the next likely requires numerous investigations
of this kind.
Teachers reading this type of account should not be overly optimistic and assume
that the
entire transformation can take place in a single lesson such as this.
Keywords: Connections, Geometry, Trigonometry
A problem from the Japanese lesson on areas of triangles in the TIMSS
study was extended to include the use of trigonometry. The problem in question
was the one in which two non-intersecting line segments are drawn, one below the
other, and the left-hand portion of the space between the segments is said to
represent "Eda's land", while the right-hand portion is "Azusa's land". The
border line between the two sides is bent, and the problem in the lesson is to
straighten the line, while preserving the areas of both parties' properties.
The class' solution was to form a triangle by drawing a "base" connecting the
ends of the bent border, then draw a line through the point in the border that
is parallel to the base, and to move the point down along the parallel line
until it touches the bottom border. The concept used in the solution was the
property that all triangles with the same base length and height have the same
area. The authors have observed this exercise u!
sed by teachers in a professional development setting, and have noted that
preliminary ideas for solutions usually involve trying to find a straight border
line that is parallel to the base of the triangle. This gave the authors the
idea to extend this lesson to involve trigonometry, and pose the two problems:
1) find the segment parallel to the base of the triangle that will preserve the
areas of the properties; and 2) find the line at some given angle to the bottom
border that will preserve the areas of the properties. Trigonometric solutions
were applied to answer both of these questions.
Although the authors did not discuss the use of these exercises in a
trigonometry class, I think they would be good extension problems that would
help connect the students' trigonometry instruction to their prior knowledge of
geometry. The exercises require a synthesis of a range of knowledge and skills
and are set in an interesting real-world context. My concern is that the
authors' trigonometric solutions appeared to be fairly complex and involved.
Perhaps these problems should only be tackled by classes at more advanced stages
of trignonometry instruction, and perhaps they should be only presented as long-
term, or even extra-credit projects.
Keywords: Problem Solving, Geometry, Algebra
The authors of this article argue that current mathematics curricula
focus too
heavily on obtaining correct solutions (emphasis on "how") and not enough on
mathematical reasoning (emphasis on "why"). Often, the only course in which
students are expected to provide explicit chains of reasoning is geometry. The
authors argue that reasoning should be promoted in all mathematics courses, and
that students should gain experience with this process prior to taking a
geometry
course. In this article they present four problems promoting exploration and
reasoning, which involve aspects of algebra and geometry, but which could be
given
to students prior to a geometry course. The problems deal with finding the
dimensions of a rectangle of fixed perimeter that maximize its area, and with
comparing areas of squares and circles with equal perimeters. A variety of
approaches are suggested to solve these problems, including exploration of
tables
and graphs (graphing calculators are recommended), use of calculators to compare
areas expressed in decimal form, and formal algebra involving formulas for
perimeter and area of rectangles and circles.
I agree with the authors that this type of exploration problem should
become an
integral part of all math curricula, not just geometry. Such problems promote
reasoning and communication skills; they lend themselves to meaningful and
appropriate uses of technology; they involve deep and interesting mathematics;
they make connections among different mathematical topics; and they help
students appreciate the necessity of building a repertoire of knowledge and
techniques in diverse areas of mathematics. Perhaps most importantly, the
regular
use of such problems helps students understand that mathematics is not just
about
"how" to solve things, but it is also about investigating "why" things happen.
Keywords: Inquiry, Connections, Technology
The author describes an exploration activity in which the problem is
posed to
students: What is the shortest possible length of cable needed to link 3 cities
that are
at specified distances apart from one another, and what is the shape of this
optimum
network? First simple cases are demonstrated to the whole class: an isosceles
triangle is shown, and students are introduced to the concept of minimum
spanning trees (paths spanning the lengths of the shortest 2 sides of a
triangle). Next
students are asked whether a different type of connection, for example, a T-type
junction (segment connecting 2 vertices, with another segment from the 3rd
vertex
to the middle of the first segment), might result in a shorter length of cable.
Finally,
students launch into the investigation, in which they attempt to discover the
type of
junction that will result in the minimum length of cable. Eventually they are
led to
the conclusion that a Steiner tree (Y-type junction with central angles of 120
degrees)
results in the optimum network (true for triangles with all angles less than 120
degrees, which is what these students are given to work with). Students move
from
exploration with paper and rulers to computer spreadsheets, in which cable
lengths
are calculated using the distance formula. The author then describes how this
activity was used in a teachers' in-service workshop. The activity was
generalized to
finding optimum networks in arbitrary triangles, and the investigation was
conducted using Sketchpad. Participants discovered that in triangles with all
angles
less than 120 degrees, Steiner trees are the optimum networks, whereas in
triangles
with an angle larger than 120 degrees, the optimum network is the minimum
spanning tree.
This activity is a great way to introduce students to graph theory, which
they
normally wouldn't see in traditional curricula. It can easily be embedded in
the
context of geometry, since it employs many geometric concepts, yet the use of
the
distance formula also provides connections to algebra. It is a rich
investigation
which could be further extended to finding optimum networks of different types
of
shapes, and it employs meaningful and appropriate use of technology. My
opinion,
however, is that the experience might be enhanced for high school students if
they
could conduct at least part of their investigation on Sketchpad. It seems an
obvious
medium for this activity, and I see no reason why high school students couldn't
use
it just as teachers in the workshop did.
Keywords: Connections, Geometry,
For her eighth-grade geometry unit, the author developed an extended 6-
week
project that draws upon the students' interests, creativity, and connections
between
geometry and other subjects. Students were offered a choice of one of three
possible
projects: 1) writing a manual with instructions and diagrams on how to do some
basic constructions using Geometer's Sketchpad; 2) creating a children's
picture
book, in which one of the main characters learns a geometry concept as part of
the
plot; and 3) writing a short report on Escher's contribution to mathematics and
creating an Escher-style tesselation. A simple rubric based on a 0-10 scale was
devised with three scoring criteria per project (shown in article; criteria
deal mainly
with aspects of the presentation rather than quality or accuracy of the
mathematical
content). The rubric was provided to students at the beginning of the unit
along
with the project descriptions. The project was assigned to four classes,
including one
for gifted students and one for learning-disabled students, and the results for
all
classes were very positive. Most students chose the picture book, but the
results of
all three projects were highly imaginative (several samples are presented).
Some
students expressed afterwards that the assignment had increased their interest
in the
subject of geometry. In my opinion this is the most important outcome of the
project. By allowing students to discover creative connections between
mathematics
and other areas of interest and talent, the subject will come alive for them.
As the
author puts it, "a good project can help us to let [their] potential shine." I
would
definitely consider assigning this type of project for a middle-school geometry
class.
Similar extended projects could also be used for high-schoolers, but the project
options must appeal to that age group - perhaps they might perceive a children's
picturebook as too babyish, for example. For the high school level, I also
believe the
rubric should contain more explicit criteria for evaluating mathematical content
than this one does.
Ref: MiriamN2
Author(s): Samide, A.J., Warfield, A.M.
Date: 1996
Title: A Mean Solution to an Old Circle Standard
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 89, May, 411-413
Reviewer: MiriamN
Date of Review: 2/5/00
This article is a case study of the phenomenon of students proposing unique
solutions to standard problems, and the interesting exercises in conjecturing,
inquiry, and proof that can arise from their solutions. The author begins by
discussing the general phenomenon and giving several reasons why such solutions
deserve some investigatory class time, including the fact that they tend to
interest
the entire class and promote mathematical connections and reasoning. The author
then launches into a specific case that occurred during a geometry class.
Students
were supposed to solve for the length of a line segment in a figure with two
externally tangent circles. The traditional solution that appears in most
textbooks
involves the construction of an auxiliary line segment and use of the
Pythagorean
theorem. However, one student came up with a completely different relationship
between the length of the segment in question and the radii of the two circles.
Students immediately became interested in the question of whether this result
was
purely coincidental or whether it would hold true in every case. The instructor
had
groups of students choose different values for measures of objects the figure
and
explore whether the student's result worked in each case. They were excited to
discover that it always did! Additional searches for counterexamples were
unsuccessful. Then students from an enriched geometry class joined this class
in
the investigation. Ultimately the students came up with two proofs of the
result,
one algebraic and one geometric. From here, students began to make further
conjectures, such as what would happen if the circles were not externally
tangent.
These questions were explored by applying algebraic and geometric methods
similar
to those they had used to do the initial proof, and they ended up discovering
interesting new relationships and generalizations. In all, several days were
devoted
to this unanticipated project.
To me, this case study is a great example of how unanticipated conjectures
that arise during class can be extended into exciting and meaningful
investigations
in which skills of creativity, mathematical connections, reasoning, and proof
can be
developed. It is also possible that original discoveries may be made, and this
is a
great way to illustrate that real mathematics is an ongoing academic pursuit
rather
than a "dead" subject for which everything is already documented in textbooks.
The
only reservation I might have about applying a similar model in my classroom is
simply the length of time that was spent on this spur-of-the-moment conjecture.
I
think it would be difficult to be that flexible with the limited time that
teachers have
to fit in all the required topics during a school year, but it would be great if
it can be
done.
Ref: MiriamN2
Author(s): Samide, A.J., Warfield, A.M.
Date: 1996
Title: A Mean Solution to an Old Circle Standard
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 89, May, 411-413
Reviewer: MiriamN
Date of Review: 2/5/00
Ref: MiriamN3
Author(s): Malloy, Carol E.
Date: 1999
Title: Perimeter and Area Through the van Hiele Model
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: 5, October, 87-90
Reviewer: MiriamN
Date of Review: 2/12/00
Ref: MiriamN4
Author(s): Pagni, David L.; Shultz, Harris S.
Date: 1999
Title: Extending a (TIMSS) Japanese Lesson Using Trigonometry
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92, March, 189-191
Reviewer: MiriamN
Date of Review: 2/15/00
Ref: MiriamN5
Author(s): Manaster, Alfred B.; Schlesinger, Beth M.
Date: 1999
Title: Geometry Problems Promoting Reasoning and Understanding
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92(2), Feb.'99, 114-116
Reviewer: MiriamN
Date of Review: 2/29/00
Ref: MiriamN6
Author(s): Iovinelli, Robert
Date: 1999
Title: Discovering Optimum Networks in Triangles
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92, Sept., 534-39
Reviewer: MiriamN
Date of Review: 3/6/00
Ref: MiriamN7
Author(s): Little, Catherine
Date: 1999
Title: Geometry Projects Linking Mathematics, Literacy, Art, and
Technology
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: 4, February, 332-335
Reviewer: MiriamN
Date of Review: 3/8/00