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 technology-based 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.

Setting of the Study

            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 four-year 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 mathematics-reasoning 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 Course Under Study

            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. 

Participants

            Fifty-two 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 twenty-three students, and the afternoon section was selected as the experimental group, with twenty-nine 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 forty-eight students, it was their first time taking the course; four students were repeating the course due to low grades or withdrawals during previous attempts.  Seventy-three (73%) percent of the participants came from public schools, and the remaining came from private institutions or other colleges, all within Puerto Rico.

Table 1

Frequencies and Percentages of Student’s Prior Grades in Mathematics

 

Control Group

(N = 23)

 

Experimental Group

 (N = 29)

A

B

C

D

 

A

B

C

D

 

5

 22%

7

  30%

 9

   39%

2

   9%

 

 

 4

   14%

 9

   31%

15

    52%

1

   3%

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.

Table 2

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 sub-topics 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 sub-topics 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 day-by-day 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.  Sub-topics 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 x-axes and (0, y) for the y-axes.

 

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 (Dufour-Janvier, 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) first-time 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 Urbana-Champaign 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 twenty-four 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 pre-test and post-test. The instrument consisted of twenty-five 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 pre-test was answered without the use of spreadsheets.  In the post-test 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.

Table 3

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 t-test and an independent samples t-test 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 t-test was used to compare prior group (control and experimental) achievement in mathematics.

 

3.      A paired samples t-test and an independent samples t-test 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 (control-experimental groups and pre-post administrations) and achievement gain in content topics and multiple representations of linear functions.

 

Summary

            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. 

 

 

[Página Anterior]