CHAPTER 3
METHODOLOGY
The focus of this chapter is the design of the study, which consisted of two parts. The first was to create instructional materials based on the use of spreadsheets supporting multiple representations of linear functions. The second was to devise an experiment to explore possible effects on student outcomes of using technologybased multiple representations. This chapter discusses the setting of the study and the subjects involved. It describes the instruments used and the data collection and analysis procedures. A schedule of activities is also provided.
This study took place at Ponce Campus of the Inter American University of Puerto Rico (IAU) during the fall semester of 2000. IAU is the largest private university in Puerto Rico with nine campuses around the Island. The Ponce Campus is a fouryear college supporting undergraduate careers in education, business, computer, natural and social sciences. Admission requirements include the College Entrance Examination Board (CEEB), administered at their schools. These standardized tests are equivalent to the SAT or ACT required at colleges and universities in the continental United States. Its maximum score is 800 points in each of the following areas: mathematics, reasoning, English, and Spanish languages. Students who score 500 or more points on this test are placed in their first mathematics course, a mathematicsreasoning course. Students with scores below 500, are placed in a basic skills mathematics course.
Among the college institutions of the area, the Ponce Campus of IAU has become one of the leaders in the use of technology. The Internet is widely used in diverse modalities, supporting distance learning courses and academic programs. The technological facilities of the Campus include a large number of computers located strategically in over five open laboratories, and at a Center of Information Access.
The mathematics course under study in this research was Mathematics Reasoning (MRSG 1010), within the Department of Science and Technology. The course meets three class hours per week and is offered every semester in several sections at various times. MRSG 1010 is a core course and is part of the general education program of the university. Since the course has a variety of instructors, a faculty member of a committee coordinates the course and its activities so that there is a similarity between sections. The course coordinator prepares a syllabus (See Appendix A), which the instructors can review and modify it, without changing the course content.
Mathematics Reasoning is a prerequisite to successive courses in the field of mathematics and science. Students whose have to take additional advanced courses in mathematics, such as precalculus and calculus, should pass it with a minimum grade of C (2.0 points in a 4.0 scale). Students registered in this course can have diverse mathematical backgrounds and levels of understanding, due to mathematics achievement location policy established by the university at the time of admission. Each instructor can choose the use of technology in this course. Several instructors have required a calculator as a course requirement.
Fiftytwo
college students registered in two sections of MRSG 1010 participated in this
study. As the result of random
assignment, the morning section was selected as the control group with
twentythree students, and the afternoon section was selected as the
experimental group, with twentynine students.
Both sections met two times per week, one hour and a half per day. The researcher was the instructor of both
groups and all the instruction was given in Spanish. The department chairman assigned the two sections to the
instructor based on availability.
Forty students were freshmen during the
semester of the study. The remaining
students were sophomores or students transferred from other colleges. The majors of the majority of the
participants were computer science (19%), biology and related fields (19%), and
business (25%). For fortyeight
students, it was their first time taking the course; four students were
repeating the course due to low grades or withdrawals during previous attempts. Seventythree (73%) percent of the participants
came from public schools, and the remaining came from private institutions or
other colleges, all within Puerto Rico.
Frequencies and Percentages of Student’s Prior
Grades in Mathematics
Control Group (N = 23) 

Experimental Group (N =
29) 








Note. A = 4.0; B = 3.0; C = 2.0, D = 1.0.
Students’ previous grades in mathematics, are summarized
in Table 1 above. As is noted in
chapter four, even though the sample means for achievement favor the control
group, this difference was found to be not significant (p > .05).
Previous experiences of participants with the
use of some kind of technology in previous mathematics courses were also
explored. Table 2 reports the
frequencies and percentages of students in, both the control and the
experimental group. These data
represent the students’ responses
(always, frequently, or occasionally) in terms of the use of technology in
their previous mathematics course. It
is noticed that more than twice as many students (proportionally) in the
control group reported experiences with calculators.
Frequencies and Percentages of Students’ Prior
Experiences with Technology
Type of technology 
Control Group 

Experimental Group 

N = 23 

N = 29 
Calculators (scientific or graphic) 
17
(74%) 

11
(38%) 
Electronic mail 
1
(4%) 

1
(3%) 
Internet 
1
(4%) 

2
(7%) 
Spreadsheets 
1
(4%) 

1
(3%) 
Other softwares (word processors, etc.) 
3
(13%) 

4
(14%) 
Treatment Description
The
length of treatment for both groups was approximately four consecutive weeks of
instruction at the beginning of the fall semester, 2000. The treatment for both
groups consisted of two parts. First,
the administration of a pre and post achievement test in linear functions, an
attitudes scale toward mathematics and a profile. Second, were the instructional activities developed in the course
content for each group. The pre and post administration of the test and the
scale took almost one week; one or two class meetings at the beginning and
another one week at the end of the study.
The treatment for the control group was based
on a traditional approach. This
approach primarily used lectures given by the instructor. The instruction paralleled the topics included in the course syllabus and the
only resource used was the textbook and handouts prepared by the
instructor. No calculators or computers
were permitted to be used by the control group. The students in this group, did however explore websites related
to the course content. All class
meetings were held in an ordinary classroom.
The experimental group received a treatment
aligned with the content topics using spreadsheets emphasizing multiple
representations as a tool to teach linear functions. Topics included in the
lessons were parallel to the course syllabus.
A computer laboratory with spreadsheet access was used for all class
meetings for this experimental group.
Due to the class size in some class meetings, students at times, had to
share computers. No more than two
students per computer were allowed. In
all class meetings, the instructor used a computer projector in order to model
the topics and activities introduced.
The
Teaching Experiment
The mathematics topics covered in the
instructional activities for both groups were the following: (a) Cartesian
coordinate system; (b) definition and graphic representation of the linear
equation, with subtopics in linear equations with two variables, solutions of
linear equations with two variables, and graphs of linear equations; (c)
intercepts, slope and the equation of the straight line, with subtopics on the
slope as a rate of change, relationship between the graph and the equation of
the straight line. The duration of each
theme was approximately one week. Due
to the nature of the mathematical content included in each of the activities,
some took more than a single day to be completed.
The Teaching Experiment with the Control Group
As
stated earlier, a traditional approach was the focus of the control group. The strategies and/or resources used during
the instructional activities were limited to instructor lectures, textbook,
handouts, some transparencies, and a Cartesian chalkboard. Following is a description of the
instructional activities developed daybyday with the control group, including
the mathematical content topics, and the proposed objectives for each
lesson. No emphasis was made with the
control group in how students moved from multiple representations.
Preliminaries for the Day One of Instruction,
Week 1
In
this class meeting, students received an orientation on the uses and
capabilities of the search engines available through the Internet. The purpose of this activity was to
encourage students to seek at least five different web sites on the content
topics included in the syllabus. These
web sites were organized by topics in order to create a database for future
reference.
Day One of Instruction, Week 2
The
main topic introduced on this class was the Cartesian coordinate system. The objectives of this lesson were to
represent points from plane in a Cartesian coordinate system and to locate
ordered pairs in a Cartesian plane.
Among the activities, students recognized the two dimensions of a
coordinate (x, y) and reviewed the concept of quadrants.
Day Two of Instruction, Week 2
During
this class, the topic taught was the definition and graphic representation of
the linear equation. Subtopics
included were linear equations with two variables and solutions of linear
equations with two variables. The
objectives of the class were the following: (a) to identify a linear equation
with two variables, (b) to determine if a given ordered pair is a solution for
a linear equation with two variables, and (c) to determine if a given point
belongs to the graph of linear equation with two variables. At this point, the form y = mx
+ b of the linear equation was introduced, where m is the slope of the line and
b is the y intercept.
Day Three of Instruction, Week 3
The
topic presented was the graph of linear equations. The main technique used was graph construction through a table of
values. Students explored the selection
of arbitrary values to assign to the x variable, the evaluation of these
values in the equation, and the finding of the corresponding y value
with different types of linear equations.
After this, they plotted and graphed the points in the Cartesian plane.
Day Four of Instruction, Week 3
At
this point of the treatment, the main topics emphasized in the instructional
lesson were the intercepts, slope and the equation of the straight line. The focus of the activities was to determine
and to describe the intercepts as well as the slope of a straight line. Students explored the line graphs with
different slopes: positive, negative, zero and indefinite. Also, students worked with intercepts on
both axes and identified the forms of an intercept: (x, 0) for the xaxes
and (0, y) for the yaxes.
Day Five of Instruction, Week 4
The
topic of this class was the slope of the rate of change and the relationship
between the graph and the equation of the line. In this lesson the formal definition of slope was introduced: _{}, and students calculated the slope given two points on the
graph. The objective of this activity
was intended to interpret the slope as a rate of change and to determine the
equation of a straight line given its slope and intercept. In addition, the formula for the general
equation of the line _{}was also used.
The Teaching Experiment with the Experimental Group
The
spreadsheets and multiple representation approach were applied to the
experimental group. This section will
describe the instructional activities developed day by day with the
experimental group using spreadsheets.
It will also include, the mathematical content covered. The objectives of the instructional
activities remained the same as the control group. For each day of instruction,
activity worksheets were distributed to students and they got printouts of
their spreadsheet work. Multiple
representations of linear functions were introduced during the class sessions
one at a time first, according to the course syllabus. Connections between representations were
established when students moved from one representation to the following one
(DufourJanvier, et al., 1987).
Preliminaries to the Day One of Instruction,
Week 1
This
class session was intended to demonstrate the capabilities of
spreadsheets. The purpose of this
activity was centered in that students get expertise and knowledge about the
capabilities of the spreadsheets. The
spreadsheet activity emphasized the use of cells, management of basic data and
evaluation of simple expressions. This
activity took one class period of instruction.
The figure below shows an example of spreadsheets.
Figure 2. Spreadsheet screen showing the use of cells and evaluation of algebraic
expressions.
Day One of Instruction, Week 2
The
topic taught was the system of Cartesian coordinates. The spreadsheet activity included the formation of the two
components of an ordered pair using cells and columns. Students plotted points on the Cartesian plane
and identified the differences when the points were moved from one quadrant to
another. Finally, students offered a
verbal description of how the application might be applied to another field.
Figure 3. Spreadsheet activity about Cartesian plane and ordered pairs.
Day Two of Instruction, Week 2
The
main topic discussed here was the definition and the graphic representation of
linear equations. The spreadsheet
activity concerned the construction of tables of values. Students determined the corresponding values
using the software capabilities. The
participants realized the effects that may have the use of different values to
construct the table. Students connected
the idea of the x and y values summarized in the tabular form
with the Cartesian coordinates located in the plane.
Figure 4. Spreadsheet activity dealing with table of values and linear graphs.
Day Three of Instruction, Week 3
The
content topic covered in this activity was the graph of linear equations. The spreadsheet was used here to construct
different tables of values and then to construct the corresponding graphs. Students observed the differences on the
graphs when they assumed different values.
The following figure gives an example of a spreadsheet in this
activity. It was emphasized in this
session about the increasing tendency observed in the table of values and the
position of the graph in the plane.
Figure 5. Spreadsheet use to teach tables of values of two linear equations and
their graphs.
Day Four of Instruction, Week 3
The
topic during this class was the slope and equation of the straight line. The spreadsheet activity emphasized the
concept of slope and linear graphs.
Students calculated the slope of a given linear equation and observed
the differences in the graph when values of m were changed; the effects of
changing the values of m in the equation
y = mx + b. It was
emphasized the four cases of slope: positive, negative, zero, and
undefined. Students described short
stories about each possible slope in linear equations. The instructor accentuated in this class the
relationship between representations and how these representations refer to the
same concept.
Figure 6. Spreadsheet screen comparing two linear graphs with different slopes.
Day Five of Instruction, Week 4
This
final class, the concept studied was the slope as a rate of change. The spreadsheet activity focused on car
prices and how them changed from year to year.
Students constructed the data table, indicating the year and the cost of
a Mercedes, and then found the value of change. Using the software capabilities, they constructed the graph. This lesson about rate of change was
designed by Burrill and Hopfensperger (1998, p. 8) and permission was granted
for its use. This class served to apply
the concept of linear functions and multiple representations to real life
situations.
Figure 7. Learning activity supported with spreadsheets about slope as a rate of
change.
Research Instruments
In
this section, there is a description of the instruments used to collect the
data for the study. The research
instruments were translated into Spanish by the researcher.
Student Profiles
In
order to collect descriptive data about participants, a profile was
administered at the very beginning of the process. This profile was designed to determine student information in the
following areas: (a) current or proposed major; (b) year of study at the time
they were taking the course; (c) previous mathematics courses passed, where
this last course was taken and the grade earned; (d) firsttime taking the
course or repeat; and (e) previous experience with technology. Sections of the profile were previously used
in a mathematics education research project conducted at the University of
Illinois at UrbanaChampaign and permission to use them was granted by the main
investigator.
Scale of Attitudes Toward Mathematics
A
scale was used for the purpose of measuring students’ attitudes toward
mathematics. This same scale was used
in the research project mentioned above with permission for its use. It was administered to the both groups at
the beginning and at the end of the
study. The instrument has twentyfour items, but five items were used to
explore attitudes or feelings toward mathematics (Items 10, 14, 16, 17,
23). The first three items of the scale
were used to explore attitudes toward technology itself and its uses. For the first two items of the scale (1 and
2), the categories used included: never, almost never, seldom, frequently, and
almost always. In the remaining items,
the categories were: strongly disagree, disagree, neutral, agree, and strongly
agree. Numbers from 1 to 5 were
assigned to each of these categories, where number 5 indicated a positive attitude. In some cases, due to negative wording of an
item, the scoring was reversed.
Achievement Test in Linear Equations
An
achievement test in linear equations was given to explore students’ mastery of
this topic of mathematics. It was
administered to both groups as pretest and posttest. The instrument consisted
of twentyfive items where the different representations of a linear equation
were included. Some items on the test
came from the Test of Graphing in Science (TOGS) (McKenzie & Padilla, 1986)
and the achievement tests samples from the Second International Mathematics
Study (SIMS) (International Association for the Evaluation of Educational
Achievement, 1995). Permission to use
was granted. The items included in the
test were grouped in the following theme areas discussed in class instruction:
Cartesian coordinates, linear equations and their graphs, and slope. The test included multiple choice items as
well as open questions.
In the experimental group, the pretest was
answered without the use of spreadsheets.
In the posttest administration, it was permitted.
Three
regular instructors (Instructor A, Instructor B, and Instructor C) of MRSG 1010
at Ponce campus of IAU evaluated the achievement test in linear functions used
in this study. They considered as
appropriate the relationship between the number of items included in the test
and the content topics taught in the course.
Nevertheless, Instructor A observed that the numbers of items by content
area are too many. He said:
The emphasis on linear equation in the course
under study is not extensive. The focus
of the course is mainly the algebraic aspect.
A large number of applications are not emphasized throughout the course.
All instructors classified some of the items as
easy and difficult for the students, based in their experience teaching the
course frequently. Table 3 reports
their comments about the test items.
Classification of Achievement Test Items Based
on the Instructors’ Responses
Instructors 
Easy Items 

Difficult Items 
Instructor A 
1, 2, 3, 4, 5, 8, 9, 10, 11, 12, 23, 24, 25 

6, 7, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22 
Instructor B 
1, 2, 3, 5, 8, 9, 10, 12, 14, 20, 23, 24, 25 

4, 6, 7, 11, 13, 15, 16, 17, 18, 19, 21, 22 
Instructor C 
1, 2, 3, 5, 9, 12, 15, 16, 17, 18, 19, 20, 22 

4, 6, 7, 8, 10, 11, 13, 14 21, 23, 24, 25 
Statistical Analysis
This
section of the chapter describes the statistical analyses done in the study in order
to answer the research questions. The
SPSS statistical software system was used for all analyses in this
project. These questions suggested the
following analyses:
1.
A paired samples
ttest and an independent samples ttest on the achievement in linear equations
test responses of the control and experimental groups comparing their
performance in the pre and post examinations and the effectiveness of the
treatment. Using factor analysis,
clusters of mathematical topics were identified in order to determine student
performance in particular areas of instruction. Two clusters on the achievement test in linear functions were
found. First, on content topics:
Cartesian coordinates system, graphs, and slope. Second, on the different
representations of linear functions: graphical, verbal, tabular, and symbolic.
2.
An independent
samples ttest was used to compare prior group (control and experimental)
achievement in mathematics.
3.
A paired samples
ttest and an independent samples ttest on the attitudes toward mathematics
responses of the control and experimental groups comparing their performance in
the pre and post examinations. The following items clusters were used in the
analysis: opinion toward mathematics, use of technology, utility of
mathematics, and study skills in mathematics.
4.
Two way analysis
of variance (ANOVA) was used to determine the significance between effects
(controlexperimental groups and prepost administrations) and achievement gain
in content topics and multiple representations of linear functions.
This
chapter has included the discussion of the methodology and the procedures
followed in this research study. The
instructional activities used with control and experimental groups were discussed
and samples about how spreadsheets were handled were also included. The research instruments that served to
collect the data from this study were described. Finally, the statistical analyses conducted in order to answer
the research questions were discussed.
The
next chapter will include the results of this research. It will discuss how participants from
control and experimental groups performed in achievement on linear equations
with and without the use of spreadsheets and their changes in attitudes toward
mathematics.