Joye Thaller
CD 145
Final Paper
May 2, 2005
The Effect of Geometer’s Sketchpad on Seventh Graders’ Understanding
of Area and Perimeter
Abstract
Area and perimeter are concepts that are usually hard for seventh graders to fully grasp.
While they may memorize formulas, they often show difficulty applying the concepts.
Past research has shown that students who construct their own knowledge tend to
understand it better and retain it longer. Geometer’s Sketchpad is a dynamic geometry
software that may aid in helping students understand more fully the concepts of area,
perimeter, and circumference. Students can make their own conjectures and create
powerful ideas using this software. This design study proposes that a group of students
taught by formal classroom instruction be compared to a group of students learning the
same material using Geometer’s Sketchpad. The students’ concept knowledge will be
evaluated with pre- and post-tests, including a post-test six months later to determine long
term effects. Student attitudes towards the material covered and to the software will be
evaluated with student surveys. The results of this study may indicate that math teachers
should use Geometer’s Sketchpad more in their classrooms in lieu of more formal
teaching methods.
1
Introduction
The concepts of perimeter and area are usually taught to children in middle
school. In the past, an instructivist approach has usually been used, where formulas and
definitions are simply given to children, who are then expected to memorize these
definitions and regurgitate them on tests. Area and perimeter can be difficult concepts
and many children have trouble understanding the relationship between the two. In this
paper, I include circumference under the broader heading of perimeter.
I believe computers may aid in students’ understanding of math. In particular, the
dynamic geometry software, The Geometer’s Sketchpad, has been used successfully by
many teachers for teaching geometry concepts in both the middle grades and in high
school geometry classes. However, few research studies have been done to determine
what the effect is of Geometer’s Sketchpad versus regular classroom instruction. I believe
this is an area in need of more in-depth research and so I propose the following research
questions for this study:
1. What are children’s pre-conceptions of area and perimeter/circumference
before instruction?
2. Does Geometer’s Sketchpad aid in children’s understanding of area and
perimeter/circumference compared to that of children taught by formal classroom
instruction?
3. What are the long term effects of using Geometer’s Sketchpad compared to
learning by formal classroom instruction?
2
Literature Review
Mindtools are “computer applications that require students to think in meaningful
ways in order to use the application to represent what they know.” (Jonassen, 2000, p. 4)
In other words, the computer should not be thinking for the child or telling the child what
to do. Vygotsky’s theories say that children use tools (or artifacts) to mediate between the
user and the environment. The computer is a material artifact, like a pencil, a calculator,
or a textbook, that acts as a mediator in the education process. It is not the educator. A
child working with a computer program should be able to go at his or her own rate,
choose the way he or she approaches the project, and see his or her own work in the
computer’s output. It is important that the type of software being used act as a Mindtool.
Dynamic geometry environments are a type of Mindtool. The term “dynamic
geometry” implies that parts of a geometric shape can be moved. Its use in a computer
environment offers the user an opportunity to develop intellectual activities based on
geometrical knowledge (Kuntz, 2002). Key features of a dynamic geometry environment
include: 1) a mathematically solid base for the implemented geometry, 2) a user-friendly
interface to allow the user to quickly learn the key facets of the software, and 3) fast
feedback to let the user have control of the objects he or she has constructed. One of the
benefits of using dynamic geometry software is that the user can freely analyze problems
in an active and dynamic way without the difficulties of trying to draw and manipulate
shapes by hand.
Papert (1980) talks about computer software as a “powerful tool” when it is used
by a child in an active and self-directed way and when the knowledge is acquired for a
recognizable personal purpose. Dynamic geometry software can act as a powerful tool to
3
identify and examine geometric properties (Santos-Trigo & Espinosa-Perez, 2002).
Students can construct, visualize, revise and change their own figures or constructions
instantaneously. Thus, students have the opportunities to make their own conjectures. The
software becomes a powerful tool in that it allows students to make and verify their own
conjectures about geometry. The conjectures may be personally meaningful to the
students, particularly if the conjecture is then used for some higher purpose (e.g. coming
up with a conjecture about how to maximize area for a garden). Sometimes, these
conjectures are existing geometric proofs. Other times, students come up with conjectures
that have never been thought of or proved before. Both can be seen as powerful ideas and
are equally important in students’ learning about geometry.
Schwartz (1989) furthers this point in his paper about computer programs as
intellectual mirrors. “Using such environments as ‘intellectual mirrors,’ it is possible for
users to probe their own understanding of a domain, as well as to devise new
relationships among the objects of a domain (Schwartz, 1989, p. 58). The software he
describes in his paper is called the Geometric Supposer. The three main features of the
Geometric Supposer that allow it to be called an intellectual mirror are: 1) it asks no
questions of the user and makes no inferences about the user’s intentions; 2) its primitive
functions in its tool menu are reasonable close to the elementary operations of the formal
system of plane geometry; and 3) it provides an environment for allowing students to
understand how the particularity of their efforts fits into a larger mathematical generality.
These characteristics are important in helping students construct their own knowledge.
There are other types of geometry programs very similar to the Geometric Supposer, such
as Cabri-Geometry and Geometer’s Sketchpad. Because it is slightly newer and is already
4
in use in many classrooms around the country, Geometer’s Sketchpad will be the focus of
this research study. This proposal does not imply that one piece of software is better than
the other. In fact, further research studies could compare different types of dynamic
geometry software. The decision to use Geometer’s Sketchpad was mainly because it was
easier for the researcher to access.
There have been few detailed studies of the effect of Geometer’s Sketchpad on
learning. One study focused on the amount of structure the teacher gave the students
when doing a Geometer’s Sketchpad activity (MacGregor & Thomas, 2002). Past
research indicated that increased structure, in the method of step-by-step instruction, can
increase learners’ perception of guidance and provide uniformity of learning outcomes
with specific outcome expectations. Less structure, however, promotes more learner
control and self-regulation, more creativity, and better meta-cognitive skills. The study by
MacGregor & Thomas was designed to determine if the instructional approach influences
students’ conceptual understanding of geometry, and also identify the students’
perceptions of the support provided by the instructor, challenges, frustrations, and
successes while working on the project-based activities. The participants were 82 tenthgraders working on a geometry transformation problem with Geometer’s Sketchpad. In
the first two groups, students were provided with specific problems to solve and given a
set of guidelines to follow. In the second two groups, students were provided with the
same problem-based learning activity, but were responsible for identifying the
information and strategies they needed to solve the problem. The method of data
collection was to have students write in an electronic diary and to write reflections in
response to questions from the teacher. The results of the study indicated that the students
5
in the teacher-directed group seemed to have a better understanding of the geometrical
concepts. The students in the self-directed groups expressed higher levels of frustration in
doing the activity. However, many students in the self-directed groups expressed a sense
of self-confidence and pleasure with their accomplishments. An important lesson to take
from this study is that the teacher has to be careful not to let students get to the point of
frustration with the activity. While it is good to let students construct their own
knowledge, the students should still be guided and provided with enough structure to
make progress and not get frustrated. The teacher selected for this proposed design study
should have a good grasp of how to provide enough structure to guide the students
without instructing them step-by-step.
Another study focused on teacher-student interaction in an activity using two
different types of geometry software (Hannafin et al., 2001). Since it is has been
established that teachers can provide instructional scaffolding in the form of questioning
and prompting, the researchers wanted to determine how teachers would provide
scaffolding in an open-ended learning environment. The researchers created an
instructivist learning environment using two computer based tools: a dynamic geometry
program (Geometer’s Sketchpad) and a computer-assisted tutorial program (Introduction
to Geometry). The overall goal of the study was to investigate the effects of the two
different types of geometry programs. Students worked in pairs on activities and also kept
journals. The students had access to both programs on their computers. The teacher’s role
was to circulate through the room and encourage the students to use the tutorial when
needed or to ask appropriate questions. Because there were so many variables in this
study, it is difficult to say whether the Geometer’s Sketchpad provided a better learning
6
environment for the students or not. The study was mostly qualitative and cited quotes
from the students such as, “I think I learned better using the Sketchpad than a normal
class because it was hands-on.” While these quotes provided some positive evidence, the
small sample size of the study (12 students) was not enough to provide conclusive results.
In addition, the focus of this study was more on teacher and student attitudes than on the
learning of a particular math topic. The research proposed in this study will focus more
on the use of the software to learn a particular topic in geometry.
The geometry topics that will be studied using Geometer’s Sketchpad are area and
perimeter of rectangles and area and circumference of circles. There are many reasons
why area and perimeter deserve attention in a research study. The Massachusetts State
Frameworks for Measurement in Grades 7-8 say students should:
Demonstrate an understanding of the concepts and apply formulas and procedures for
determining measures, including those of area and perimeter/circumference of
parallelograms, trapezoids, and circles. Given the formulas, determine the surface area
and volume of rectangular prisms, cylinders, and spheres. Use technology as appropriate.
(Massachusetts State Frameworks, 2001)
The NCTM Principles and Standards for Mathematics for Grades 6-8 say that
students should “develop and use formulas to determine the circumference of circles and
the area of triangles, parallelograms, trapezoids and circles and develop strategies to find
the area of more-complex shapes.” (NCTM, 2004, p. 240) In particular, NCTM says that:
“Whenever possible, students should develop formulas and procedures meaningfully
through investigation rather than memorize them. Even formulas that are difficult to
justify rigorously in the middle grades, such as that for the area of a circle, should be
treated in ways that help students develop an intuitive sense of their reasonableness.”
(NCTM, 2004, p. 243)
7
Area and perimeter are not always easy topics for students, which is why NCTM
advocates investigation rather than rote memorization. Boston and Smith (2003) say that
“situations in which one attribute of a geometric figure is allowed to vary while another
attribute remains constant can provide rich-problem solving opportunities.” (p. 208) In
one vignette, Boston and Smith (2003) cite a case where three seventh-graders argued
vociferously that a fixed amount of rabbit fencing would always form the same area no
matter which shape you put it in. After the teacher asked them to try it on their calculator
instead of just guessing, they were convinced that the calculator results were wrong. It
was only after going into the hallway, roping off a perimeter, and counting the square
tiles on the floor, did they come to the realization that their original theory was wrong
and that the same perimeter could result in different areas. Clearly, this was a case of
Piagetian disequilibrium. Fortunately, the teacher realized this and kept asking them
questions until they saw for themselves that their original conjecture was incorrect.
In a study by Woodward and Byrd (1983), 129 eighth-graders were given a task
to find the garden with the largest area out of six differently-shaped rectangles, all with a
perimeter of 60 meters. Only 23% answered the question correctly (the square garden is
the largest), while 59% of the students said all the gardens were the same size. Students
from a mathematics course for prospective elementary school teachers were asked the
same question. Almost two-thirds of the teachers stated that the gardens were all the same
size. Although the students and teachers had learned the formula for calculating area and
the dimensions of each garden were given, somehow the connection was not made that
gardens with the same perimeter were not necessarily the same size. Clearly, something
was lacking in the way in which area was taught to these people. A formula-based
8
approach using only computational techniques does not work. Formulas should instead be
developed inductively.
In my own experience in a seventh-grade classroom, I have seen many children
(working on a similar problem) state that the volume of different cylinders of various
heights will be the same if the surface area is the same (it is not, the maximum volume is
the one whose radius equals its height). However, it is not until the children actually fill
these containers with beans and measure the volume for themselves that they see that the
volume varies.
Geometer’s Sketchpad allows students to “see for themselves” in a virtual
environment. Since they control what the computer is doing, they will know it is not
lying when it shows two rectangles of the same perimeter having different areas. They
can also develop formulas inductively and test them with the software. Figure 1 shows an
example of the way in which Geometer’s Sketchpad can be used to explore rectangles,
while Figure 2 shows an exploration of circles. In this proposed design study, students
will use Geometer’s Sketchpad to explore area and perimeter of rectangles and area and
circumference of circles.
9
Figure 1. Example of Geometer’s Sketchpad used to explore area and perimeter of rectangles.
Figure 2. Example of Geometer’s Sketchpad used to explore circumference of circles.
10
Methodology
The research proposed for this study will be both quantitative and qualitative. The
pre-test and post-tests will be analyzed quantitatively, while the videotapes and surveys
will provide qualitative data. Portions of the survey will also be assessed quantitatively.
Sample
The sample will consist of two classes of seventh graders (20-25 students in each
class) taught by the same teacher, at a school to be determined. The lesson will span two
days: one day on rectangles and one day on circles. It is important that the students not
have formal instruction in the geometry of rectangles or circles yet, so the study will be
done at the beginning of the school year. The teacher will teach one class with
instructivist methods without using computers. The lesson can be taken out of a textbook
or other source of information that the teacher would like to use. The other class will use
computers with Geometer’s Sketchpad to do the activity. The students will work in pairs
on the activity. The teacher will guide the students through the activity, providing help
when needed and asking provoking questions to stimulate critical and complex thinking.
The teacher should not specifically teach the students in the computer group, but should
help when needed so that students do not get too frustrated. Each worksheet is broken
into two smaller exercises. For the first exercise, one student should use the computer
while the other student records the information. For the second exercise, the students will
switch roles. On the second day, the students will follow the same routine.
11
Data Collection
Prior to the lesson, all of the students will be given the same pre-test (Appendix
A). This test will determine their prior knowledge of rectangles and circles, before any
instruction has been given. During the lessons (for both the computer and non-computer
group), video documentation will be taken to assess student participation, collaboration,
discourse, etc. The students in the computer group will complete the activity worksheets
shown in Appendix B, while the teacher may choose to assign or not to assign any
worksheets for the instructional group.
After the two day lesson, students will be given a post-test (Appendix C), which
contains similar concepts but different questions than the pre-test to determine the
amount knowledge the students have gained from the lesson. The students will all be
given a written survey (Appendix D) to document their attitudes about the lesson. The
students will not be interviewed individually for this part, mainly due to time constraints
and because there will also be video documentation of the classes.
Approximately six months after the lesson, the students will be given another
post-test (Appendix A). This test will be exactly the same as the pre-test. The students
will presumably not remember the exact questions (or answers) to the pre-test, but will
hopefully remember some of the concepts. For this reason, pre-test will be recycled
instead of creating a third test.
12
Data Analysis
The pre-test and post-tests will be scored by the researcher. The multiple choice
items will be given a value of 0 or 1, with 1 being the correct answer. The open response
questions will be scored on a point system, according to the following rubric:
0
Question is
left blank or
answer is
irrelevant to
question
1
Student shows
little to no
understanding of
concept and
shows little or no
work.
2
Student shows some
understanding and
makes attempt to solve
with numbers. Student
uses or writes down
correct formula.
3
Student shows some
understanding of
concept. Student
uses or writes down
correct formula.
Answer is partially
correct.
4
Student
demonstrates full
understanding of
concept and
provides correct
answer.
The multiple choice and open response scores will be added to give a total score for each
student. The average of the scores for the computer group will be compared to the
average of scores of the non-computer group to see if there is any statistical difference for
each of the three tests. The three tests will also be compared to each other within each
group to see if there is any statistical difference on the tests before the lesson, after the
lesson, and six months after the lesson.
The videotapes will record students’ engagement with the software or in the
instructional lesson. A researcher will walk around with the camera to record the
interaction in the computer pairs to see how well they are working together and to record
images from the computer screen itself. Since the teacher cannot help every student at
once, the videotapes will show if students were using the software correctly, what
problems they encountered, and how much they explored on their own.
The surveys will be evaluated according to student responses on the 5-point Likert
scale. The values will be averaged for the computer group and the non-computer group to
assess student level of enjoyment and confidence for each group. The purpose of the
13
extra questions for the computer group is to assess whether the students enjoyed using
Geometer’s Sketchpad and to determine whether it should be used for future lessons. It is
possible that the computer group enjoyed the subject of geometry, but did not enjoy the
specific computer program, so it is important that these questions be on the survey.
The teacher will also be interviewed to see whether he/she enjoyed teaching this
lesson with the software and would want to use it again. It is assumed that the teacher
chosen will have prior experience with Geometer’s Sketchpad or will be taught how to
use it prior to the lesson. It is important that the teacher be comfortable using the software
because it is likely that the teacher’s attitude will affect the students’ attitude.
Implications and Future Research
The results of the study may show that students using Geometer’s Sketchpad
learned more and retained more knowledge than the students who were taught with the
instructional method. If this is the case, then a case can be made to include more dynamic
geometry software in seventh grade classrooms. If there is no difference or if the
instructional group scores higher, than the videotapes should be examined further to
determine if there were problems the students were having that prevented them from
learning as much as they could have.
Future studies should be done on a larger sample of students with more than one
teacher to validate the results of this study. In addition, other types of dynamic geometry
software, such as the Geometric Supposer and Cabri-Geometry, should be studied and
compared to Geometer’s Sketchpad to see if one is easier to use or results in more student
learning.
14
The results of this study will hopefully push for more technological tools for use
in math classrooms, so that students can create their own geometry instead of being
taught it. As students learn to form conjectures at a seventh grade level, they will learn to
think in critical, creative, and complex ways. Those students with previous knowledge in
formal reasoning and conjecture making will have an advantage when they get to high
school geometry, where they will have to develop and understand formal proofs.
Geometer’s Sketchpad will send them in the right direction.
15
References
Boston, M. & Smith, M. (2003). Providing Opportunities for Students and Teachers to
Measure Up. In D.H. Clements & G. Bright (Eds.), Learning and Teaching
Measurement. (pp. 208-219). Reston, VA: National Council of Teachers of Mathematics.
Hannafin, R., Burruss, J., and C. Little. (2001). Learning with Dynamic Geometry
Programs: Perspectives of Teachers and Learners. The Journal of Educational Research,
94(3), 132-150.
Jackiw, N. (1995). The Geometer’s Sketchpad. Berkeley, CA: Key Curriculum Press.
Software.
Jonassen, D.H. (2000). What are Mindtools? In D.H. Jonassen & D.A. Stollenwerk
(Eds.), Computers as mindtools for schools: Engaging critical thinking (pp. 3-20). NJ:
Prentice Hall.
Kuntz, G. (2002). Dynamic Geometry on WWW. In Borwein, J., Morales, M., Polthier,
K., & J. Rodrigues (Eds.), Multimedia Tools for Communicating Mathematics (pp. 221229). Berlin: Springer.
MacGregor, S. & Thomas, R. (2002, June). Learning Geometry Dynamically: Teaching
Structure of Facilitation?
Paper presented at the National Educational Computing
Conference, San Antonio, TX.
16
Massachusetts Department of Education. (2001). Massachusetts mathematics curriculum
framework. Malden, MA: Massachusetts Department of Education.
National Council of Teachers of Mathematics (2000). Principles and Standards for
School Mathematics. Reston, VA
Papert, S. (1980). Computers and computer cultures. Mindstorms: Children, computers,
and powerful ideas (pp. 19-37). NY: Basic Books.
Santos-Trigo, M. & Espinosa-Perez, H. (2002). Searching and exploring properties of
geometric configurations using dynamic software. International Journal of Mathematical
Education in Science & Technology. 33(1), 37-50.
Schwartz, J. (1989). Intellectual Mirrors: A Step in the Direction of Making Schools
Knowledge Making Places. Harvard Educational Review, 59, 51-60.
Woodard, E. & Byrd, F. (1983). Area: Included Topic, Neglected Concept. School
Science and Mathematics, 83(4), 343-346.
17
Appendix A
Pre-Test (and 6-month Post-Test)
1.
Field A
30 meters
2 meters
Field B
16 meters
16 meters
Which field will require more fertilizer? (circle one)
A. Field A
B. Field B
C. They both require the same amount
2. Imagine a string wrapped around the earth so that it touches end to end. Now supposed
that string were raised on six foot high poles all the way around the earth. How much
more string would be needed so that the ends could still touch?
10 ft
10 ft
3. Jane wants to make a rabbit pen for her rabbits and has 200 meters of fencing to
enclose it. She wants to make sure the rabbits have as much room as possible to move
around. Draw below what you think the rabbit pen should look like. Label all sides.
A-1
Appendix B
Activity Worksheets
Relationships between Area and Perimeter
Objective: To achieve a deeper understanding of the concepts of area and perimeter and
their relationships.
Question 1: For quadrilaterals with the same perimeter, what shape will have the largest
area?
Conjecture: Write a statement that you think accurately answers the question above.
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
Use the table below to investigate:
Relationship between Area and Perimeter of a Quadrilateral: Part 1
Height
Width
Perimeter
Area
Question 2: For quadrilaterals with the same area, what shape will have the largest
perimeter?
Conjecture: Write a statement that you think accurately answers the question above.
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
Use the table below to investigate:
Relationship between Area and Perimeter of a Quadrilateral: Part 2
Height
Width
Perimeter
Area
Can you write a generalization based on your findings?
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
A-2
Relationships between Area and Circumference
Objective: To achieve a deeper understanding of the relationships between radius, area
and circumference in circles.
Questions: As you change the radius of a circle, how does its area change? How does its
circumference change?
Conjecture: Write a statement that you think accurately answers the questions above.
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
Radius
Relationship between Radius and Circumference in a Circle
Diameter
Circumference
Circumference/Diameter
Do you notice anything about the result when you divide the circumference by the
diameter? What is this number? ______________________________________________
Can you come up with a formula to calculate the circumference of a circle using its
radius?
Circumference =
Radius
Relationship between Radius and Area in a Circle
(Radius)2
Area
Area/(Radius)2
Do you notice anything about the result when you divide the area by the radius squared?
________________________________________________________________________
Can you come up with a formula to calculate the area of a circle using its radius?
Area =
A-3
Appendix C
Immediate Post-Test
1. The following boxes all have the same area. Which box has the largest perimeter?
A
B
C
2. Calculate the area and the perimeter of the square below.
6 cm
3. What is the radius of a circle whose area is 100 cm2?
4. What is the circumference of a circle whose diameter is 10 cm?
5. The earth’s radius is 6400 km. What is its circumference? If the earth expanded and its
radius increase to 6410, what would its new circumference be? What would be the
change in circumference?
A-4
Appendix D – Survey
1. On a scale of 1 to 5, with 1 hating it and 5 loving it, what are your feelings about math?
2. On a scale of 1 to 5, with 1 the easiest and 5 the hardest, how hard did you think the
lesson on area and perimeter of a quadrilateral was?
3. On a scale of 1 to 5, how much did you like the lesson on quadrilaterals?
4. On a scale of 1 to 5, with 1 the easiest and 5 the hardest, how hard did you think the
lesson on circles was?
5. On a scale of 1 to 5, how much did you like the lesson on circles?
4. How confident are you in your ability to do math? (1 – no confidence, 5 – very
confident)
5. How confident are you in your ability to use computers? (1 – no confidence, 5 – very
confident)
(just for the group using Geometer’s Sketchpad)
7. What did you like about Geometer’s Sketchpad?
8. What did you not like?
9. Would you like to use the program for future lessons?
A-5