The Use of Computer Software in High School Geometry

Al Bierschbach

Dakota State University

The Use of Computer Software in High School Geometry

Introduction:

Computer use in the teaching and learning of mathematics is being encouraged and research shows that it is beneficial (Manoucherhri, 1999). By using computer technology, open-ended learning environments are created in which learners become active managers, rather than passive receptacles, of information (Hannafin and Scott, 1998).  This study will explore the following question: Are students more interested in Geometry when they use dynamic software such as The Geometer's Sketchpad?

Review of the Literature:

The Geometer's Sketchpad can be helpful in exploring geometric patterns and relationships (Bonsangue, 1997).  The interactive software allows students to manipulate geometric figures to develop and formulate postulates and theorems, and develop geometric constructions (Alba, 1998).  Geometric constructions can enrich students' visualization and comprehension of geometry and allow students to apply their creativity to mathematics (Sanders, 1998).  The Geometer 's Sketchpad can create a rich mathematical environment that students can explore and promote a conjecturing spirit that is fundamental for advancement in math and lifelong learning (Manouchehri, Enderson, and Pugnucco, 1998). 

In a study by Dixon (1997), she showed that students could be more successful in Geometry with an increased knowledge of computer software, such as The Geometer 's Sketchpad.  In a study outline by Pokay and Tayeh, students who were studied using The Geometer 's Sketchpad felt that when first using sketchpad acknowledged that they were uncomfortable with the software, frustrated with the language and did not like that the computer did not simply give them the answers.  With increased use of the software, the students used the software as a tool and were much less frustrated (Pokay and Tayeh, 1997).  In this study, student's interests in Geometry will be noted while using The Geometer 's Sketchpad. 

Method:

Subjects:  The subjects of this study are fourteen Geometry students from

Madison High School in Madison, South Dakota.  The class contains twelve sophomores and two juniors.  The two juniors completed Pre-Algebra as freshman and Algebra I as sophomores.  The sophomores all completed Algebra I as freshman.  The subjects have various ability levels and have had computer experience in other classes, but have not had the experience of using The Geometer 's Sketchpad software program. 

Focus:  The focus of this study is the subject's observable attitudes and comments about Geometry while using The Geometer 's Sketchpad. 

Setting:  Two different classrooms at Madison High School.  One classroom is normally used for English classes and the other classroom is a computer lab with enough computers so each subject will be at a computer. 

Role of the Observer:  The observer is a participant role, as he is also the teacher of the students in this Geometry class. 

Type of Data Collected:  The data collected will be the comments, both positive and negative, about Geometry and the use of The Geometer's Sketchpad.  Visible observations of the students being on-task or off-task will also be noted. 

Analysis:

To explore the functions of The Geometer's Sketchpad, the students followed Tour 1, Tour 2, and Tour 3 from the guidebook for sketchpad.  These tours are good for the students to explore different functions of the software without me having to lecture the students through all of the functions.  Students do not always follow directions precisely, and when one student misses one step, the others get off-task while the one student is assisted to catch up to the rest of the group. 

The goals of the lesson were given in the English classroom.  The students were told how to access the software on the computers and what some of the basic controls looked like.  It was also explained how to change one tool (such as the line tool) to another tool (such as the ray tool).  The students were told to follow the tours to learn the basic functions of the software. 

While working with the software, Cale did not follow the first tour, opting to play and explore with the different functions.  Ann followed the tours and went on to make different polygons then the tour asked for.  She also commented that, "This is cool." Jackie, who does not follow my directions all that well, said that what we were doing was, "Boring." She did not follow tours 2 and 3.  Matt, who had very little success in Geometry so far this year, said, "This is kind of cool." Cory, who can get very frustrated very easily, made the comment, "1 did it."

In this first exposure to Sketchpad, not all of the students showed excitement

about learning with the software.  One of the advantages of using the software is that the students experience geometry in a different way other than the teacher lecturing or modeling.  The next day after the first lab experience, Heather, Jackie, and Ann asked when they would get to go back to the computer lab.  With comments like that, it could show that the students were interested in the software.  Jackie even asked the next week when the next trip to the computer lab was scheduled, which shows someone who thought the first experience was "boring", could be interested with the software. 

The next trip to the computer lab occurred the day after Jackie's last question of when the class would get to go to the lab next.  After discussing interior and exterior angles of polygons the day before, the class went to the computer lab to see if one of the theorems could be demonstrated.  The goal was to create a pentagon whose exterior angle sum was 360 degrees, as was given by a theorem on the day before.  The students were expected to make a pentagon using the ray tool and they had to measure each exterior angle of the pentagon, which after going through the correct steps, was displayed on the computer screen.  Then they had to print their results. 

John, who is very comfortable with computers, constructed the polygon perfectly the first time and went one step further with the measurements and used one of the program's functions to display the sum of the exterior angles, which was 360 degrees.  After he showed me how to display the sum of the angles, he also showed Cale, Grant and others how to display it.  Some of the students did not understand why their angles did not add up to 360 degrees, and after a little investigation, it was discovered that they did not name their angles in the correct order with the vertex as the middle letter.  This made the computer measure a different angle than what the students wanted.  Jackie, who thought the computer program was boring during the first session of using it, followed the directions very well and did an excellent job.  She did not have any negative comments this time, which can be noted as an improvement. 

With helping others, this showed an interest for others to share their knowledge of the software to enhance everyone's learning.  The absence of negative comments displays an appreciation for the software.  After two sessions, there are already positive effects that are coming from the use of the software. 

The next week the class went to the lab to practice what we had learned the day before, the ASA postulate.  The students were to demonstrate that given two angles and an included side of one triangle are congruent to the corresponding angles and side of another triangle, and then the two triangles are congruent.  All of the students played with the triangles and found when they moved the angles or side, the triangles were still congruent.  Then they looked at a SSA (Side-Side-Angle) situation, and showed how two triangles could be constructed with two consecutive congruent sides and a non-included angle.  They first discovered how the triangles were congruent, but then they constructed them so they were not congruent, which proved that SSA was not a triangle congruence theorem or postulate.  Cale said, "I understand why SSA doesn't work."

Most of the students were more confident in their abilities and did not ask many questions, they just went ahead and did their work until they found non-congruent triangles.  The students were on-task, which can show that they are comfortable with the software and appreciate its power to discover why Geometry has certain theorems and why certain conjectures, like SSA, will not work. 

Conclusion:

As noted earlier in this study the amount of negative comments about the software decreased and the students in the class gained an appreciation of the software.  This was just a study of the comments and actions of the students while in a computer lab using The Geometer's Sketchpad.  Due to limitations on time and access to the computer lab, only three separate observations could be made in this initial study. 

Further Development of the Research:

This study needs to continue to collect comments while using the software, along with comments of students in the normal class setting, to see if the software does actually impact the students' interest level in Geometry. Other observations that could be made are smiles on the students' faces as compared to frowns.  Data should be collected to see how comments about Geometry change when working through proofs of Geometry by paper and pencil compared to using Sketchpad to work through a proof.  Data should be collected over more than one group of students to see if there are any differences in the comments. 

References:

Bronner, E.  (1998, September 30).  Computers help math learning, study finds.  New York Times, p. 20.

 Ginsburg, A.  (1998, October 21).  Test scores and computer use:  Longitudinal studies needed.  Education Week, 18, 38.

 Manoucherhri, A.  (1999).  Computer and school mathematics reform:  Implications for mathematics teacher education.  The Journal of Computers in Mathematics and Science Teaching, 18, 31-48.

 Geometry [computer software].  (1997).  Intelligent Software, Inc.

 Pokay, P. and Tayeh, C. (1997).  Integrating technology in a geometry classroom:  Issues for teaching.   Computers in the Schools.  14, 117-123.

 Roberts, D. L., & Stephens, L. J. (1999). The effect of the frequency of usage of
            computer software in high school geometry .The Journal of Computers in
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