Chapter IV

RESULTS

    This chapter presents the results obtained during the investigation. The study was designed to investigate the effect of a computer-based curriculum on mathematical reasoning and problem solving skills at the college level. First, there is a description of the results obtained from the pre and post assessment. Second, there is a report of the results from the self assessment instruments and the interviews.

The Pre and Post Tests

    Results of the study are reported as an analysis of the pretest results, posttest results and the difference between them. Table 4 presents the achievement pretest means, posttest means, standard deviations, and minimum and maximum values for the three groups. The preliminary analysis shows that the pretest mean of the Computer 1 group was higher than that of the Computer 2 or the Non-Computer Groups. This analysis shows also that the Non-Computer Group scored considerable higher when compared to the pretest mean of the group and considerably higher when compared to the mean of the other groups for the posttest scores. Table 5 gives the results of the t-test analysis in relation to achievement mean gain scores. There was significant difference (p<.05) in achievement mean gain scores in the Non-Computer group. There was no significant difference in achievement mean gain scores when the mathematical reasoning course was taught with the use of computers.

Table 4

Means and Standard Deviations of the Pre-and-Post Testing


Computer 1

Mean 
SD
Minimum
Maximum
Pre 
10.76
5.00
0
20
Post
12.36
6.10 
0
22

Computer 2

Test 
Mean 
SD 
Minimum
Maximum
Pre 
5.64
4.47
1
17
Post 
7.00
5.74
0
17

 Non-Computer

Test 
Mean 
SD
Minimum
Maximum
Pre 
7.42
5.15
0
16
Post 
16.21
4.23
6
24





Table 5

Summary of the t-test Analysis


Group
Test
Mean
SD
p

Computer 1 (C1)
Pre
10.76
5.00
 
 
Post
12.36
6.10
.252
Computer 2 (C2)
Pre
5.64
4.47
 
 
Post
7.00
5.74
.202
Non-Computer (NC)
Pre
7.42
5.15
 
 
Post
16.21
4.23
.000*


*p2<. 05.

    A more detailed analysis of the processes involved in problem solving of the individual examinations might provide some better information on the differences between the groups. Table 6, 7, 8 and 9 show the number of students that selected a strategy that could have led to a correct solution if implemented properly to solve problems in the pre and post tests.
The most frequent process used for problem 1 (Table 6) in the pretest was the strategy Make a Table. This characteristic applies to the three groups. One student from the Computer 1 group and five students from the Non-Computer group selected a strategy that could have led them to the correct solution of problem 1 in the Posttest.

Table 6

Selection of Strategies for Problem 1 in the Pre and Post Tests


Strategy
C 1
C 2 
PP
 
N=25
N=25
N=24

Find a pattern 
1(0) 
0(0) 
1(4)
Draw a diagram 
 0(0) 
0(0) 
0(0) 
Guess and check 
 0(0) 
0(0) 
0(0) 
Use an equation 
 0(0) 
0(0) 
0(0) 
Make a table 
  17(l) 
18(0) 
16(l)


Note. Values enclosed in parentheses represent posttest selections. C1 = Computer 1 group; C2 = Computer 2 group;
NC = Non-Computer group.
The results for problem 2 in each test were interesting (see table 7). More than fifty percent of the students in each group selected an appropriate strategy. For these problems the strategy most often used was Make a Diagram.
Table 8 shows an interesting shift which occurred in the performance of the three groups in problem 3 of each test. For problem 3 in the pretest, three students (4%) selected an appropriate strategy, compared to fifty four students (73%) in the posttest. The Computer 1 and the Non-Computer groups scored similarly in these problems.

Table 7

Selection of Strategies for Problem 2 in the Pre and Post Tests


Strategy
C 1
C 2
PP
N=25
N=25
N=24

Find a  pattern
0 (0)
0 (0)
0 (4)
Draw a  diagram 
13 (16)
14 (10)
17 (15)
Guess and check 
0 (0)
0 (0)
0 (0)
Use an  equation
0 (0)
0 (0)
0 (0)
Make a  table 
1 (0)
0 (0)
0 (0)


Note. Values enclosed in parentheses represent posttest selections. CI = Computer 1 group; C2 = Computer 2 group;
NC = Non-Computer group.
 

Table 8

Selection of Strategies for Problem 3 in the Pre and Post Tests


Strategy
C I 
 C 2 
PP
 
N=25 
N=25 
N=24

Find a pattern
0(0)
0(0)
0(0)
Draw a diagram
0(0)
0(0)
0(0)
Guess and check 
0(20)
0(11)
0(15)
Use an equation 
0(0)
0(0)
0(0)
Make a table 
1(0)
0(3)
2(5)


Note. Values enclosed in parentheses represent posttest selections. C1 = Computer 1 group; C2 = Computer 2 group;
NC = Non-Computer group.

    Table 9 shows not only an interesting shift in the performance of the groups, but a difference in the selection of the strategies. The Computer 1 and the Computer 2 groups showed a strong preference for the strategies Guess and Check and the Make a Table, while the Non-Computer group showed a preference for the strategy Use an Equation. Although both groups received instruction on the use of equations to solve problems, students in the computer groups applied strategies similar to those used with the computer.

Table 9

Selection of Strategies for Problem 4 in the Pre and Post Tests


Strategy
C 1
C 2
PP
 
N=25
N=25
N=24

Find a pattern 
0(0)
0(0)
0(0)
Draw a diagram
0(0)
0(0)
0(0)
Guess and check
4(23)
6(8)
4(7)
Use an equation 
0(1)
2(7)
2(17)
Make a table
3(1)
0(2)
2(0)


Note. Values enclosed in parentheses represent posttest selections. C1 = Computer 1 group; C2 = Computer 2 group;
NC = Non-Computer group.

Part Two in each test was designed to assess students' qualitative reactions to their performance in each problem. After they solved each problem, students were asked to answer a set of questions (see Appendix A). Table 10 summarizes the percentage of "yes" responses to questions 1, 2 and 3 of Part Two in each test.

Table 10

"Yes" Responses to Questions 1, 2 and 3 of Part Two of the Pre and Post Tests


                                                                                               % Responding "yes"

Question 
C1 
C2 
PP

       1:Seen before?
16(29) 
17(27) 
9(26)
       2:Seen related? 
40(62)
34(38)
43(56)
3:Know how to start? 
46(57)
42(33) 
 44(57)

Note. Values enclosed in parentheses represent posttest responses.

    At the time of the pretest the groups' perception about their knowledge was comparable, with a slight difference in the Non-Computer group about problems they had seen before. The Non-Computer group thought that they had seen more "closely-related" problems than the computer groups. The percentage of "related problems" the students reported having worked with increased for all groups in the posttest. Computer 1 group reported the highest score, followed by the Non-Computer group and the Computer 2 group.

    Students' responses for questions 4, 5 and 6 are given in Table 11. In the posttest there is an increase in the responses of the students who answered "yes" to question 4. For question 5, sixty percent of the students from the computer groups chose the categories of "organized a bit" or "organized", while in the Non-Computer group sixty-eight percent of the students chose these categories.

    There was a considerable decrease in the percent of responses in question 6 for the selection of "difficult" in the Computer 1 and Non-Computer groups. In the three groups there was an increase for the selection of "moderately difficult."

Table 11

Responses to Questions 4, 5 and 6 of Part II of the Pre and Post Tests


Question
C1 
C2
PP

4:  Did you make a plan?

No
26(17)
11(20)
8(3)
I planned a bit
49(43)
40(27)
34(39)
Yes
24(31)
27(30)
33(39)

5:  How do you feel was your work on the problem?

Disoorganized
30(14) 
 20(19) 
15(13)
organized a bit 
51(56) 
40(34)
 38(35)
organized
15(11)
20(18) 
 20(33)

6:  Please rate the difficulty of the problem.

Easy  13(16)  17(10) 16(21)
Moderately Difficult  39(55) 26(34) 35(45)
Difficult  45(18) 29(38) 34(16)

Note. Values enclosed in  parentheses represent posttest responses.

    The investigator observed the use of calculators and computers in the posttest. Ten students (40%) in the treatment group C1 used computers during the posttest. No students used computers in the treatment group C2. The students in the three groups brought calculators with them and used them for computation on a regular basis during the examination.
In the treatment groups the investigator reported that 20 hours were devoted to conference or classroom activities, 8 hours to the teaching of computer use, 12 hours to computer demonstrations and 5 hours to testing. The comparison group did not spend time learning to use a computer, but rather used this time for working in the course's content.

    The investigator reviewed the scores of the pre and post test of students who knew how to use Lotus 1-2-3 before the treatment, or had the ability and good disposition to learn to use the computer. Table 12 presents four examples of student accomplishment worthy of note. Students 3 and 26 knew how to use Lotus 1-2-3 and students 5 and 6 had the ability and a good disposition to learn to use the computer. These factors are relevant in the discussion of the results.

Table 12

 Examples of the Results in the Pre and Post Tests


Student 
 Pretest
Posttest
Difference

003
12
22
10
026
5
17
12
005
6
21
15
006
7
19
12

The Self Assessment Instrument

Student Attitude Questionnaire

    The investigator used the Student Attitude Questionnaire to assess problem solving attitudes. The instrument was administered at the beginning and at the end of the treatment. Table 13 presents the means and standard deviations of the scores in the Student Attitude Questionnaire. The analysis of variance (Table 14) showed that there was no significant difference between the groups at the beginning of the study. There is significant difference in two of the dimensions of the questionnaire, self-confidence and perseverance, at the end of the study. Scheffe test with significance level .05 indicated that the differences are found in the Computer 1 and Non-Computer groups. These results suggest that the computer is not necessarily the reason for change in the attitudes of both groups.

Table 13

Means and Standard Deviations of the Scores in the Student Attitude Questionnaire


GROUP 
SCALE
MEAN
SD
 
 
PRE
POST
PRE
POST

 
Self-confidence
2.30
2.95
1.14
0.91
Computer 1 (C1)
Perseverance
2.30
2.95
1.14
0.91
 
Willingness
3.33
3.58
0.92
0.90

 
Self-confidence
2.74
2.17
0.59
0.99
Computer 2 (C2)
Perseverance
2.74
2.17
0.59
0.99
 
Willingness
3.59
3.06
0.89
1.00

 
Self-confidence
2.29
2.95
1.15
0.84
Non-Computer (NC)
Perseverance
2.29
2.95
1.15
0.84
 
Willingness
3.67
3.55
0.91
0.91

 

Table 14

Analysis of Variance Between Groups for the Student Attitude Questionnaire



 
Scale
Pre
Post

F
F

Self-confidence
.1700
.0136*
Willingness
.3979
.1697
Perseverance
.1700
 .0136*


*P<.05
Fennema-Sherman Mathematics Attitudes Scales

    The scales Usefulness of Mathematics and Confidence in Learning Mathematics were administered to the subjects at the beginning and at the end of the study. The t-test analysis results (Table 15 and 16) showed that there is no significant difference in the attitude of the subjects in the computer groups toward the usefulness of mathematics These results suggest that the use of technology does not necessarily change students' attitude in these dimensions.

    Similar results were found with the sub-scale Confidence in Learning Mathematics. There is no significant difference in the students' confidence to learn mathematics in the computer groups.

Table 15

Results of the t-test Analysis in the
Fennema-Sherman Mathematics Attitude Scale

    Confidence in Learning Mathematics


Test
Mean
SD
P
Pretest
2.88
.346
 
     
.460
Posttest
2.98
.395
 


 

Table 16

Results of the t-test Analysis in the
Fennema-Sherman Mathematics Attitude Scale



                        Usefulness of Mathematics

Test
Mean
SD
P
Pretest
2.93
.352
 
     
.759
Posttest
2.94
.416
 

    The results of the t-test analysis in the Fennema-Sherman Mathematics Attitude Scale demonstrated that the students had a positive attitude toward the usefulness of Mathematics or confidence in learning mathematics. Tables 17 and 18 show that for positive statements in the Scales, students selected "strongly agree" and "agree."

Table 17

Mean of the Combined Percents of Strongly Agree and Agree
in the Fennema-Sherman Mathematics Attitude Scales


Usefulness of Mathematics Scale

                    Statement                                                                     Mean

1. I'll need mathematics for my future work.
84.8%
2. I study mathematics because I know how useful it is.
88.2%
3. Knowing mathematics will help me earn a living.
71.6%
4. Mathematics is a worthwhile and necessary subject.
94.8%
5. I'll need a firm mastery of mathematics for my future work.
65.3%
6. I will use mathematics in many ways as an adult.
95.0%

Table 18

Mean of the Combined Percents of Strongly Agree and Agree in the Fennema-Sherman Mathematics Attitude Scales



Confidence in Learning Mathematics Scale

                 Statement                                                                     Mean

1. Generally I have felt secure about attempting mathematics.
67.0%
2. I am sure I could do advanced work in mathematics.
65.1%
3. I am sure that I can learn mathematics.
93.6%
4. I think I could handle more difficult mathematics.
53.3%
5. I can get good grades in mathematics. 
78.0%
6. I have a lot of self-confidence when it to mathematics.
63.5%


Perception Toward the Computer Scale

  The Perception Toward the Computer Scale was administered to the three groups. Interesting results were found between the Computer groups and the Non-Computer group in some of the dimensions of the scale (Tables 19, 20 and 21). The Computer groups scored a higher percent than the Non-Computer group when asked about the usefulness and effectiveness of  computers. Other areas where the Computer groups scored higher were in the characteristics of amusing, interesting, necessary and powerful. In general students had similar perceptions in the characteristic of simple (difficult) and pleasant (unpleasant).

Table 19

Percents of the Results of the Perception Toward the
Computer Scale in the Computer 1 Group (N =22)


 
-
-
-
0
+
+
+
 
 
3
2
1
 
1
2
3
 

Useless
0
0
0
0
5
18
77
Useful
Ineffective
5
0
5
5
0
17
68
Effective
Unnecessary
0
0
0
14
5
22
59
Necessary
Uneasy
0
0
0
14
14
27
45
Easy
Difficult
0
0
18
27
14
18
23
Simple
Unpleasant
0
0
9
5
5
36
45
Pleasant
Boring
0
0
0
9
9
18
64
Amusing
Tedious
0
0
5
9
9
13
64
Interesting
Powerless
0
0
5
13
0
18
64
Powerful

Table 20

Percents of the Results of the Perception Toward the
Computer Scale in the Computer 2 Group (N = 21)


 
-
-
-
 
+
+
+
 
 
3
2
1
0
1
2
3
 

Useless
0
5
0
5
5
5
80
Useful
Ineffective
5
0
5
10
0
14
66
Effective
Unnecessary
5
0
5
0
9
5
76
Necessary
Uneasy
9
0
0
0
19
19
57
Easy
Difficult
9
9
5
19
19
29
10
Simple
Unpleasant
5
5
5
5
5
19
57
Pleasant
Boring
5
5
5
0
0
24
61
Amusing
Tedious
5
5
5
0
0
5
80
Interesting
Powerless
5
5
0
5
5
0
80
Powerful

Table 21

Percents of the Results of the Perception Toward the
Computer Scale in the Non-Computer Group  (N = 21)


 
-
-
-
 
+
+
+
 
 
3
2
1
0
1
2
3
 

Useless
5
0
0
0
16
21
58
Useful
Ineffective
0
0
0
5
11
37
47
Effective
Unnecessary
5
0
0
5
11
26
53
Necessary
Uneasy
10
0
0
5
16
32
47
Easy
Difficult
0
0
16
16
21
32
42
Simple
Unpleasant
0
0
0
5
21
32
42
Pleasant
Boring
0
0
5
0
21
26
42
Amusing
Tedious
0
0
5
0
11
42
42
Interesting
Powerless
0
0
0
5
11
37
47
Powerful

The Interviews

    The investigator interviewed thirteen students during the last week of the study. See the interview protocol in Appendix L. The results of the interviews relates to the students' perception toward computers. When asked about their experience in the course, 76% of the students said it was "good." Other answers to this question were "I liked it"; "Interesting"; "Difficult"; "Different"; and "Awful."

Some of their comments were "the course expanded my knowledge about the importance of mathematics," "the course helped me to learn more about mathematics" and "the course helped me to learn about computers." One students said "the
course made me use my imagination and taught me to be patient."  These responses were encouraging since this was the first experience the students had in using a computers in a mathematics class. Other suggestions given by the students to improve the use of the computers in the course were "to have a tutor or an assistant during the laboratory periods," "I would like to be in a computerized classroom all the time," and "I would like more practice time (He meant more practice time during the period of time they were in class.)."

    To the question "Do you think that the computer is a useful tool when solving problems?", 92% of the students answered "yes." All of them agreed that they would like to use computers in other mathematics courses. To improve the use of the computers in the course, students (77%) said that they would rather prefer to take the Introduction of Computers course before using computers in a mathematics course.

    Although the investigator developed computer-based activities for all the topics in the course, the students had some preferences about when to use it. Table 22 shows students' responses to the question "In what topics of the course is the computer more suitable?."

Table 22

Students' Preferences of the Topics Where Computers Should
Be Used


CHAPTER
%

Chapter 1- Problem Solving (Inductive Reasoning, Patterns)
11
0.85
Chapter 2- Linear Equations in One Variable
6
0.46
Chapter 3- Ratio and Proportion, Percent  
8
0.62
Chapter 7- Statistics
12
0.92
Chapter 5- Linear Equations in Two variable
8
0.62


 
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