Chapter III


    This chapter discusses the subjects of the study and the instruments used as well as a description of the environment and the instructional materials.

Subjects of the Study
    The study was carried out in a private four-year university in Ponce, Puerto Rico, during the Spring of 1995. The students varied in age from 18 to 45; thirty seven percent (37%) of the students were males, sixty three percent (63%) were females. Most students were from the southern region of Puerto Rico. The sample distribution is shown in Table 1.
The data analysis for the pre and posttest was restricted to 74 students because six students did not take the pretest, and 25 did not take the posttest. Table 2 shows the number of subjects in each treatment of the experiment.
The Course
    One of the components of the core curriculum in the university where this study was conducted is Reasoning Skills. Bachelor's degree candidates are required to enroll

Table 1

Sample Distribution


No major 75.0%
Education (B.A.)/Elementary (General)    5.0%
Computer Science (B.S.)    3.4%
Accounting (B.B.A.)   2.6%
Secretarial Science (B.A.)   1.7%
Business Administration (A.A.S.)   1.7%
Education Secondary/English as a Second Language    1.7%
Psychology (B.A.)/General    1.7%
Criminal Justice (B.A.)/Criminal Investigation   0.9%
Management (B.B.A.)    0.9%
Nursing (A.A.S.)    0.9%
Office Information System    0.9%
Biology (B.S.)/Pre-Medical    0.9%
Managerial Economics (B.B.A.)    0.9%
Biology (B.S.)/General    0.9%
Business Administration (B.A.)    0.9%


Table 2

Participating Groups

                                                 Group                                                             Size

Note. C denotes the group that used  computers, NC denotes the group that used paper and pencil.
for eight credits in this area, including the following courses: Logical and Critical Reasoning [LRSG 1010, 3 credits], Introduction to Computers [CRSG 2010, 2 credits], and Mathematical Reasoning [MRSG 1010, 3 credits]. The Mathematical Reasoning Skill Course was selected because it related to the purpose of the study. The University catalog (1995-97) describes the course as follows:

Use of estimation, interpretation of graphs, solution of equations, financial management and statistical concepts to promote mathematical thinking and the development of heuristic strategies for problem solving. Study of the following concepts: linear
equations in one and two variables, systems of linear equations and their graphs, annuities, amortization, graph representation of numerical data, measures of central tendency and dispersion. Use of the calculator as a principal working tool. (p. 104)
    A copy of the outline for Mathematical Skill Course appears in Appendix F. The major topics covered in the course are: The Nature of Patterns and Inductive Reasoning [two weeks], Linear Equations [three weeks], The Nature of Statistics [three weeks], The Nature of Financial Management [three weeks], and The Nature of Graphs and Systems [four weeks].

    Although the Mathematical Reasoning Skill Course was developed to promote mathematical reasoning and the development of problem solving skills, it was first taught as a means to provide students with the algebraic skills they needed for other college courses. Later, an institutional committee revised the description of the course and modified it to fulfil its original purpose.

    Some topics were chosen to provide algebraic skills, while others, were chosen because of their application to daily life situations. The course intends to improve students' problem solving and mathematical reasoning skills. To accomplish this goal the course's content includes the use of the Polya's method (Polya, 1985) for problem solving and its application to various topics or situations, specially non-text-type problems. Both roles--problem solver and problem poser--are practiced by the professors.

The Environment
    The subjects of the study were students registered in MRSG 1010, Mathematical Reasoning, a first-year college level course. The register divided the students into three groups. Two groups were designated Computer 1 (C1, Instructor A), and Computer 2 (C2, Instructor B). The third group was designated Non-Computer (NC, Instructor A). The only planned difference in the treatments was that the Computer groups worked problems involving the concepts and skills from the mathematical reasoning course using the computer; while the Non-Computer group did not use the computer in the study of the same mathematical content. The treatments groups and the comparison group used the calculator for computational purposes. The investigator taught Computer 1 and Non-Computer groups. A mathematics department colleague taught group Computer 2.

    The three groups met three times per week during a 15-week academic semester. The investigator and the colleague working in the project scheduled conferences for the treatments groups C1 and C2 in a regular classroom and demonstrations in the computer assisted classroom [CAC]. The comparison group NC met in a regular classroom during the whole semester.
During conference time the professors worked on the course's content emphasizing the use of the Polya's method (Polya, 1985), which consists of four basic steps of the problem-solving process. The steps are Understand the Problem, Make a Plan, Carry Out the Plan and Look Back. At this time, both roles--problem solver and problem poser-were practiced by the professors. As a problem solver the intention was not to present the solution to a problem in a neat and clean presentation. Schoenfeld's work (1987), presented in Chapter II, suggested that this polished presentation often obscures the process, giving the impression that things should be easy for people who study mathematics. As a consequence the participants saw a "problem resolution" not a "problem solution."

    The students in the treatments and comparison groups received instruction on problem solving strategies. These strategies were: Find a Pattern, Guess and Check, Make a Table and Draw a Diagram, Use and Equation. For the purpose
of this project, the five problem solving strategies mentioned can be defined as follows:

    Another way of helping the participants in the problem solving process was to moderate students' activity. The professors asked questions to give the opportunity to the students to assess their own progress. Other activities in the problem solving process were observation, guidance of students' individual or small group work, and discussion of the solutions attempt.
In the computer assisted classroom, professors demonstrated to the students how they could use computers to solve problems. The C1 and C2 groups received three sessions of instruction in the use of the spreadsheet Lotus 1-2-3. Additional instruction was given as needed for the purpose of the project.
Instructional Materials
    During the experiments groups C1, C2 and NC used the textbook Fundamentos de Razonamiento Matemático (Smith, 1991). The textbook presents a problem solving approach, following the Polya's (Polya, 1985) method for problem solving. The groups used a booklet of selected activities for the course. The activities of the treatment groups C1, C2 differ from the activities of the comparison group NC in terms of the format. The format for the activities of the treatment groups C1, C2 included suggestion of the software to be used and examples of spreadsheets. The format for the activities of the comparison group NC included only the problems. See Appendix G for the activities used by the treatment groups, Selected Activities for the Mathematical Reasoning Skill Course, Part 1. Additional activities appear in Appendix H, Selected Activities for the Mathematical Reasoning Skill Course, Part 11. The treatment groups C1, C2 received permission to photocopy an instructional booklet used to teach the spreadsheet Lotus 1-2-3.

    The objective of the activities used with the students was to experience the use of tables to find a pattern, the use of tables to make a guess, or the use of a graph that could lead to the problem solution. These strategies for solving problems were worked with a spreadsheet to take advantage of the computational functions built-in this kind of software. Spreadsheets are considered especially appropriate in activities involving quantitative data analysis (Brown, 1986-87). The spreadsheet allows students to see a progression of calculations on the screen all at once, thus facilitating them to see patterns develop. Like the calculator, the spreadsheet not only allows students to experiment with numbers and develop numerical concepts, but also allows teachers to choose more realistic problems without concern for about extremely difficult calculations.  Although the advantage a student can find in using a spreadsheet, it is important that students should be concentrating on the strategy, not the computer.
Both booklets contain activities for all the topics discussed in the course. The topics included in the activities were Patterns and Inductive Reasoning, Linear Equations, Percents and Problem Solving, Financial Management, Graphs, and Systems. The booklets contained fifty activities, and a collection of twenty four non-routine problems. Due to the limited amount of time available, twenty five of these activities were used during the experiment.

The Instruments
The Pre and Post Tests

    The mathematical reasoning and problem solving skill tests (Appendix B) were developed to assess achievement of mathematical and problem solving skills of students who received instruction on heuristic strategies in a one semester course. The instruments were used as the pretest and posttest of the study. The purpose of the instruments was the assessment of mathematical reasoning and problem solving skills, not the mathematics involved. The instruments were paper and pencil tests, but since they were developed to examine the effects of computer use on mathematical reasoning and problem solving skills at the college level, students in the treatment groups had the opportunity to use computers during examination periods. All the problems in the tests could be solved with the aid of a calculator. Students in the treatment groups had the opportunity to decide between both tools. Students in the comparison group used the calculator.

    The two measures are equivalent, not equal. The strategy that can be used to solve a pretest problem can also help to solve a posttest problem. Table 3 presents specifications of the strategies suggested. To evaluate students' tests, the investigator and the colleague working in the study used the analytic scoring scale suggested by Charles, Lester, and O'Daffer (1987). Appendix I contains the scale. Problems in the test may be solved by the chosen strategy, although the problems could be solved using other strategies. A t-test analysis was applied to the results of both measures.

    After each problem in both test, the students responded to a qualitative assessment of their performance on problem solving. This measure gives information about the experience of the students with the type of problem work on the test.

Table 3

Specifications for the Pre and Post Tests

Problem Solving Strategy 
Type of Problem
Find a pattern 
Draw a diagram 
Guess and check 
Use and equation 


The Self Assessment Instruments

Student Attitude Questionnaire

    The Student Attitude Questionnaire, SAQ, (Moses, 1976) is a group-administered paper-pencil inventory to assess problem solving attitudes. This instrument was selected because the scale contains the dimensions of attitudes that were of interest to the investigator. The dimensions are willingness to engage in problem solving activities, perseverance during the problem solving process, and self-confidence with respect to problem solving. A copy of the questionnaire is included in Appendix D.
SAQ was developed at Indiana University as a part of a National Science Foundation sponsored project on problem solving.

    The permission to use the instrument was granted by Professor Frank Lester from Indiana University. The instrument has a coefficient alpha reliability of .79.

    The scales Willingness and Perseverance consist of six items each. The scale Self-confidence consists of eight items. The three scales consist of positively and negatively stated items. The scale Willingness includes items such as "It is not fun to try to solve problems" and "I like to try hard problems." The scale Perseverance includes items such as "I give up on problems right away" and "I will work a long time on a problem." The scale Self-confidence includes items such as "I can only do problems everyone else can do" and "I can solve most hard problems."

    The instrument was translated from English into Spanish by the investigator. To validate the Spanish version, an English specialist made a back translation into English. The back translation confirmed the original translation. The Spanish translation was submitted to a jury of five to establish content validity and to adapt the instrument for the college level.

Fennema-Sherman Mathematics Attitudes Scales

    The Fennema-Sherman Mathematics Attitudes Scales (1976) consists of nine scales: Confidence in Learning Mathematics Scale, Usefulness of Mathematics, Attitudes Toward Success in Mathematics Scale, Mathematics as a Male Domain Scale, Father Scale, Mother Scale, Teacher Scale, Mathematics Anxiety Scale, and Effectance Motivation in Mathematics Scale.  The investigator selected the scales Confidence in Learning Mathematics and Usefulness of Mathematics. Appendix C contains a copy of the scales.

    The instrument was already translated into Spanish by a colleague in the Mathematics Department and used in her doctoral dissertation (Nolasco, 1988). The coefficient alpha reliability of the instrument is .98. Permission to use the Spanish version was granted by Dr. Nolasco.

Perceptions Toward the Computer Scale

    The students' attitudes toward the computers were assessed using Part IV of the scale Percepciones Hacia la Tecnologia. The instrument was developed by Mrs. Brunilda Figueroa in the Office of Planning, Development and Evaluation in the university where the study was conducted. Three main types of computer attitudes were measured: liking computers or enjoying working with computers, usefulness of computers, and comfort with computers. The instrument has a coefficient alpha reliability of .94. Appendix E contains a copy of the subscale.

The Interviews

    Interviews were conducted with a selected sample of students from the treatments groups. The purpose of the interview was to get students' thoughts, understanding and feelings about the use of computers in the Mathematical Reasoning Skill Course. Appendix J presents the protocol for the interviews.

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