Chapter III

METHODOLOGY

    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 Course

Table 1
 

Sample Distribution

Table 2

Participating Groups

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)

    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 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:

•  Find a Pattern

  The strategy consists of identifying a pattern in given information or data, and then making a generalization that can  be tested. Exploring patterns and arriving at generalizations is the heart of the strategy.
 
• Guess and Check

 This strategy is based on the fact that once an initial guess is made, the student has a point of reference from which refinements can be made. By checking a guess the student learns how to make a better guess. Following this process the student can find the answer.
 
• Make a Table

 This strategy consists of making a systematic table of the known information. This method provides an opportunity to identify the information and record data in a manner suitable for examination. From such examination simple conclusions can often be reached.
 
•  Draw a Diagram

This strategy consists of drawing a diagram or a graph. This method provide the student a semi-concrete representation of the problem which is appropriate for examination. From such examination the student can find the solution of the problem.
 
• Use and Equation

 The student translates the verbal expression to an equation. Solving the equation the student can obtain the solution of the problem.

Instructional Materials

    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 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

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.

    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.

    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.

    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.
 

[Pág. Anterior]