CHAPTER 4
RESULTS
The goal of this study was to create mathematics lessons based on the use of spreadsheets emphasizing multiple representations of linear functions. It also aimed to explore possible effects of instructional uses of multiple representations on students’ outcomes (attitudes and achievement).
This chapter presents the findings of this study. The results are described in the following sections: Prior achievement in mathematics based on grades; Attitudes toward mathematics; and Achievement in mathematics (linear functions).
Prior Achievement in Mathematics Based on Grades
These data were obtained through the students’ profiles administered to the control and experimental groups at the beginning of the study. In order to compare the performance in mathematics between the groups under study, an independent samples tTest was carried out. Table 4 reports the results.
Prior Mathematics of Control and Experimental Groups
Variable 
Group 
M* 
N 
SD 
t 
df 
P 
Prior Achievement 
Control 
2.65 
23 
0.93 
0.42 
50 
0.68 
(Based on Reported Grades) 
Experimental 
2.55 
29 
0.78 



Note. *Higher mean, better prior achievement in mathematics.
Table 4 reveals that no significant (p > .05) was found in this comparison. Hence, it may be concluded that at the beginning of the project, the experimental and the control groups were comparable in prior mathematics achievement, based on reported grades.
Attitudes Toward Mathematics
The attitudes toward mathematics that control and experimental groups exhibited at the beginning and end of the study were also explored. The items were divided into the two clusters: students’ opinion or feelings about mathematics (5 items) and attitudes toward technology (1 item) and its use (2 items). For these items, the following four statistical comparisons among the groups were made: precontrol vs. preexperimental; postcontrol vs. postexperimental; prepost control; and prepost experimental.
Table 5 and Table 6 show the results of the first and second comparison described above, respectively.
Precontrol and Preexperimental Groups’ Attitudes Toward Mathematics
Variable 
Group 
M* 
N 
SD 
t 
df 
P 
Attitudes Toward 
Control 
17.35 
23 
2.87 
1.06 
50 
.293 
Mathematics 
Experimental 
16.28 
29 
4.10 



Note. * Higher mean, positive attitudes toward mathematics.
In Table 5, a ttest reveals that the difference between the groups in attitudes toward mathematics was not significant (p > .05) at the beginning of the study.
The second comparison on attitudes toward mathematics carried out in this study was the postcontrol vs. postexperimental. Table 6 reports the results of this analysis.
Postcontrol and Postexperimental Groups’ Attitudes Toward Mathematics
Variable 
Group 
M* 
N 
SD 
t 
df 
P 
Attitudes Toward 
Control 
16.70 
23 
3.23 
.006 
50 
.995 
Mathematics 
Experimental 
16.69 
29 
3.73 



Note. * Higher mean, positive attitudes toward mathematics.
The comparison on attitudes toward mathematics between control and experimental groups at the end of the study indicates no significant difference (p > .05).
In order to compare means between pre and post administration of the attitudes scale toward mathematics in control and experimental groups, the statistic analysis carried out was a paired samples ttest. Tables 7 and 8 summarize the results of the comparison between pre and post administration of the attitudes scale toward mathematics in the control and experimental groups, respectively.
Pre and Post Attitudes Toward Mathematics: Control Group
Variable 
Group 
M* 
N 
SD 
t 
df 
P 
Attitudes Toward 
Precontrol 
17.35 
23 
2.87 
.95 
22 
.353 
Mathematics 
Postcontrol 
16.70 
23 
3.23 



Note. * Higher mean, positive attitudes toward mathematics.
A t statistic reveals that this difference is not significant (p > .05)
Table 8 below reports the comparison between pre and post administration on attitudes toward mathematics in the experimental group.
Pre and Post Attitudes Toward Mathematics: Experimental Group
Variable 
Group 
M* 
N 
SD 
t 
df 
P 
Attitudes Toward 
Preexperimental 
16.28 
29 
4.10 
.652 
28 
.520 
Mathematics 
Postexperimental 
16.69 
29 
3.73 



Note. * Higher mean, positive attitudes toward mathematics.
Similarly, for these comparisons, no significant difference (p > .05) was found between the groups in attitudes toward mathematics.
The following table summarizes the distribution of frequencies and percentages on students’ responses on the five items dealing with attitudes or feelings toward mathematics
Distribution of Frequencies and Percentages on Students’ Responses on the Five Items Dealing with Attitudes Toward Mathematics
Item 
Group* 
N 
SD** 
D** 
N** 
A** 
SA** 
10. It scares me to have to take mathematics 
PreC PostC PreE PostE 
23 23 29 29 
2.0 (8.7) 5.0 (21.7) 7.0 (24.1) 4.0 (13.8) 
9.0 (39.1) 7.0 (30.4) 4.0 (13.8) 9.0 (31.0) 
11.0 (47.8) 9.0 (39.1) 9.0 (31.0) 14.0 (48.2) 
1.0 (4.3) 1.0 (4.3) 7.0 (24.1) 1.0 (3.4) 
0.0 (0.0) 1.0 (4.3) 2.0 (6.9) 1.0 (3.4) 
14. I am looking forward to taking more
mathematics. 
PreC 
23 
0.0 (0.0) 
4.0 (17.4) 
12.0 (52.2) 
6.0 (26.1) 
1.0 (4.3) 
14. I am looking forward to taking more
mathematics. 
PostC PreE PostE 
23 29 29 
2.0 (8.7) 3.0 (10.3) 2.0 (6.9) 
4.0 (17.4) 3.0 (10.3) 5.0 (17.2) 
8.0 (34.8) 14.0 (48.3) 10.0 (34.5) 
9.0 (39.1) 7.0 (24.1) 11.0 (37.9) 
0.0 (0.0) 2.0 (6.9) 1.0 (3.4) 
16. No matter how hard I try, I still do not do well
in mathematics. 
PreC PostC PreE PostE 
23 23 29 29 
9.0 (39.1) 5.0 (21.7) 7.0 (24.1) 2.0 (6.9) 
8.0 (34.8) 9.0 (39.1) 10.0 (34.5) 11.0 (37.9) 
3.0 (13.0) 6.0 (26.1) 5.0 (17.2) 12.0 (41.4) 
2.0 (8.7) 3.0 (13.0) 6.0 (20.7) 2.0 (6.9) 
1.0 (4.3) 0.0 (0.0) 1.0 (3.4) 2.0 (6.9) 
17. Mathematics is harder for me than for most
persons. 
PreC PostC PreE PostE 
23 23 29 29 
5.0 (21.7) 4.0 (17.4) 3.0 (10.3) 3.0 (10.3) 
11.0 (47.8) 7.0 (30.4) 9.0 (31.0) 8.0 (27.6) 
5.0 (21.7) 8.0 (34.8) 8.0 (27.6) 12.0 (41.4) 
1.0 (4.3) 4.0 (17.4) 5.0 (17.2) 5.0 (17.2) 
1.0 (4.3) 0.0 (0.0) 4.0 (13.8) 1.0 (3.4) 
23. If I had my choice, this would be my last
mathematics course. 
PreC PostC 
23 23 
4.0 (17.4) 2.0 (8.7) 
8.0 (34.8) 7.0 (30.4) 
5.0 (21.7) 4.0 (17.4) 
5.0 (21.7) 8.0 (34.8) 
1.0 (4.3) 2.0 (8.7) 
23. If I had my choice, this would be my last
mathematics course. 
PreE PostE 
29 29 
3.0 (10.3) 5.0 (17.2) 
11.0 (37.9) 8.0 (27.6) 
9.0 (31.0) 5.0 (17.2) 
2.0 (6.9) 6.0 (20.7) 
4.0 (13.8) 5.0 (17.2) 
Note. *Groups: PreC = PreControl, PostC = Postcontrol,
PreE = Preexperimental, PostE = Postexperimental. **SD = Strongly disagree,
D = Disagree, N = Neutral, A = Agree, SA = Strongly Agree.
Students’ Attitudes Toward Uses of Technology
It was also important to this study to explore the differences between groups in the different areas into which the attitudes scale items were divided: use of technology (2 items) and attitudes toward technology (1 item). Table 10 reports the frequencies and the percentages on students’ responses on scale items 1 and 2 dealing with use of technology, particularly in the use of calculators in their last two years of high school.
Distribution of Frequencies and Percentages on Students’ Responses on Scale Items 1 and 2 Dealing with Technology
Item 
Group* 
N 
N** 
AN** 
S** 
F** 
AA** 
In the mathematics classes I took in the last two
years of high school, I used a calculator to perform routine calculations. 
PreC PreE 
23 29 
4.0 (17.4) 7.0 (24.1) 
6.0 (26.1) 6.0 (20.7) 
7.0 (30.4) 12.0 (41.4) 
4.0 (17.4) 2.0 (6.9) 
2.0 (8.7) 2.0 (6.9) 
In the mathematics classes I took in the last two
years of high school, I used a graphing calculator to graph functions. 
PreC PreE 
23 29 
13.0 (56.5) 23.0 (79.3) 
4.0 (17.4) 2.0 (6.9) 
3.0 (13.0) 2.0 (6.9) 
1.0 (4.3) 2.0 (6.9) 
2.0 (8.7) 0.0 (0.0) 
Note. *Groups: PreC = Precontrol, PostC = Postcontrol,
PreE = Preexperimental, PostE = Postexperimental.**N = Never, AN = Almost
Never, S = Seldom, F = Frequently, AA = Almost Always.
Table 11 contains the descriptive data about the two items of the attitudes scale dealing with uses of technology. It compares the students’ reported use of technology in the control and experimental groups.
Statistic 
Groups 
Item 1 
Item 2 
NMMdnSD R*
Q** 
Precontrol 
23 2.74 3.00 1.21 4.00 2.00 
23 1.91 1.00 1.31 4.00 2.00 
NMMdnSD R*
Q** 
Preexperimental 
29 2.52 3.00 1.15 4.00 1.50 
29 1.41 1.00 0.91 3.00 0.00 
Note. *Range = higher score – lowest score; **Interquartile range.
Table 12 reports the ttest comparison
between the control and experimental groups on the two items of the scale
dealing with uses of technology.
Precontrol and Preexperimental Groups’ Uses of Technology
Variable 
Group 
M* 
N 
SD 
t 
df 
P 
Uses of 
Precontrol 
4.65 
23 
2.17 
1.37 
50 
.176 
Technology 
Preexperimental 
3.93 
29 
1.62 



Note. * Higher mean, positive attitudes toward mathematics.
Table 12 reveals that differences in reported usage of calculators
in high school were not significant (p > .05).
The item number 3 from the scale explored the students’ attitudes toward technology. Table 13 presents the distribution of frequencies and percentages on students’ responses on item 3 from the attitudes scale toward mathematics dealing with technology.
Distribution of Frequencies and Percentages on Students’ Responses on Scale Item 3 Dealing with Technology
Item 
Group* 
N 
SD** 
D** 
N** 
A** 
SA** 
In order for me to learn mathematics, using a
calculator or computer is helpful. 
PreC PostC PreE PostE 
23 23 29 29 
0.0 (0.0) 2.0 (8.7) 3.0 (10.3) 1.0 (3.4) 
2.0 (8.7) 1.0 (4.3) 3.0 (10.3) 0.0 (0.0) 
10.0 (43.5) 10.0 (43.5) 13.0 (44.8) 8.0 (27.6) 
8.0 (34.8) 3.0 (13.0) 3.0 (10.3) 13.0 (44.8) 
3.0 (13.0) 7.0 (30.4) 7.0 (24.1) 7.0 (24.1) 
Note. *Groups: PreC = Precontrol, PostC = Postcontrol, PreE =
Preexperimental, PostE = Postexperimental.
**SD = Strongly Disagree, D = Disagree, N = Neutral, A = Agree, SA =
Strongly Agree.
However, since there is only one item in this category, there are questions as to the reliability of this measure. Therefore, box plots were used to describe graphically student performance on this item dealing with attitudes toward technology. It can be noted that the median responses on this item remained at 3.0 for the control group, but rose from 3.0 to 4.0 for the experimental group.
Figure 8 presents box plots showing the distribution of scores for the control group (pre and post) and for the experimental group (pre and post) on the item of the scale dealing with technology. It compares the changes in attitudes in both groups.
Figure 8. Box plots showing distribution of student attitudes toward technology.

Summary
This section of the chapter presented the findings on attitudes toward mathematics, divided into two major areas explored in this study: opinion or feelings toward mathematics, technology and its uses. The statistical analysis revealed no significant differences (p > .05) on the attitudes measures between the control group and the experimental group. In the experimental group, it appears, the distribution of scores (using box plots and tables above) that the attitudes exhibited toward the use of technology increased somewhat and the end of the treatment. It suggests that students in the experimental group felt more confident in the use of computers as an invaluable tool in their mathematics class.
Achievement in Mathematics
Achievement in mathematics, particularly on linear functions, was also studied. In order to obtain these data, a test with twentyfive items was administered to both groups at the beginning and end of the study. The items of this test were divided in two different clusters. Cluster A includes the items dealing with content topics taught during the treatment: Cartesian coordinates (5 items), graphs (7 items), and slope (13 items). Cluster B includes the items of the test dealing with multiple representations of the linear functions: symbolic (3 items), graphical (10 items), tabular (3 items), and verbal (9 items) representations. The following four statistical comparisons between the groups were made: precontrol vs. preexperimental; postcontrol vs. postexperimental; prepost control; and prepost experimental. In order to examine possible interactions effects between occasions (pre and post tests) and conditions (control vs. experimental groups), twoway analysis of variance (ANOVA) was also carried out. Only significant interactions are reported. Tables 14 and 15 show the results of the first and second comparisons, respectively.
Precontrol and Preexperimental Groups’ Mathematics Achievement (Linear Functions)
Variable 
Group 
M* 
N 
SD 
t 
df 
P 
Achievement in 
Control 
8.70 
23 
3.40 
4.42 
50 
.000 
Mathematics (Linear functions) 
Experimental 
4.31 
29 
3.67 



Note. *Higher mean, better achievement.
Table 14 suggests a higher achievement in mathematics in favor of the control group at the beginning of the study. The difference between means of the control and experimental groups was about 4 points. That is, the mean of the control group is almost twice the mean of the experimental group. The resulting t statistic reveals that this difference in achievement in linear functions between groups was significant (p < .05). The second comparison on achievement in mathematics carried out in this study was the postcontrol vs. postexperimental. Table 15 reports the results of this analysis.
Postcontrol and Postexperimental Groups’ Mathematics Achievement (Linear Functions)
Variable 
Group 
M* 
N 
SD 
t 
df 
P 
Achievement in 
Control 
9.70 
23 
5.03 
.17 
50 
.864 
Mathematics (Linear functions) 
Experimental 
9.93 
29 
4.78 



Note. *Higher mean, better achievement.
Interestingly, the comparison on achievement between the control and the experimental groups at the end of the study was not significant (p > .05).
Tables 16 and 17 summarize the results of the comparison between prepost administration of the test in the control and experimental groups, respectively.
Pre and Post Mathematics Achievement (Linear Functions): Control Group
Variable 
Group 
M* 
N 
SD 
t 
df 
P 
Achievement in 
Precontrol 
8.70 
23 
3.40 
1.16 
22 
.258 
Mathematics (Linear functions) 
Postcontrol 
9.70 
23 
5.03 



Note. Higher mean, better achievement.
The ttest shows that the difference in the means of the achievement in mathematics on the control group was not significant (p > .05)
Pre and Post Mathematics Achievement (Linear Functions): Experimental Group
Variable 
Group 
M* 
N 
SD 
t 
df 
P 
Achievement in 
Preexperimental 
4.31 
29 
3.67 
6.70 
28 
.000 
Mathematics (Linear functions) 
Postexperimental 
9.93 
29 
4.78 



Note. *Higher mean, better achievement.
Table 17 above, reports the comparison between pre and post administration on achievement in mathematics in the experimental group.
It is observed in this analysis that in the posttest in achievement in mathematics, particularly in linear functions, the mean of the experimental group had a dramatic increase. According to the t statistic, this difference in almost 6 points is significant (p < .05). In contrast to the control group, where achievement in mathematics increased slightly, the experimental group showed a considerable improvement in achievement in linear functions.
This trend in the data (apparent dramatically different changes in means for the control and experimental groups from the pre to the post test) was ‘unpacked’ in later analyses using twoway ANOVA (See Appendix F). The first subinvestigation deals with the clusters of items on the achievement test by content areas (Cluster A). The second subinvestigation deals with multiple representations of linear functions (Cluster B). The analyses carried out through the independent samples ttest and the paired samples ttest to both clusters revealed that there were significant differences (p < .05) in some areas. Also, using ANOVA, it was found that there were significant interactions (p < .05) between occasions and conditions for certain of the clusters. The following sections include a discussion of these areas.
Slope
Slope
constitutes another important concept in the study of linear functions. The same four comparisons between groups,
described above, were carried out in this section of cluster A. Significant difference (p < .05)
was found in only two comparisons: precontrol vs. preexperimental and
prepost experimental. Tables 18 and 19
show the results of these analyses.
Precontrol and Preexperimental Groups’ Achievement in Linear Functions (Slope)
Variable 
Group 
M* 
N 
SD 
t 
df 
P 
Achievement in 
Precontrol 
5.00 
23 
2.30 
3.79 
50 
.000 
Mathematics (Slope) 
Preexperimental 
2.43 
29 
2.50 



Note. *Higher mean, better achievement.
It
is observed that at the beginning of the study, the control group exhibited a
better achievement on slope. The
considerable difference on means between groups was about 2.55 points. The resulting t statistic indicates that
this difference was significant
(p < .05).
Table
19 reports the findings on the prepost comparison in the experimental group.
Pre and Post Experimental Group’s Achievement
in Linear Functions (Slope)
Variable 
Group 
M* 
N 
SD 
t 
df 
P 
Achievement in 
Preexperimental 
2.45 
29 
2.50 
4.65 
28 
.000 
Mathematics (Slope) 
Postexperimental 
4.83 
29 
3.31 



Note. *Higher mean, better achievement.
This
result suggests that experimental group had an improvement in achievement in
mathematics, particularly in the concept of slope. The t statistic indicates that this difference between prepost
administrations was significant (p < .05). Although this same comparison in the control group was not
statistically significant, there was a reduction on means between pre and
post. That is, the mean declined from
5.00 at the beginning of the study to 4.61 at the end of the treatment, a
difference about .39 points. Meanwhile,
in the reported comparison on the experimental group, the difference between
means reached approximately 2.38 points.
Table
20 reports the results of the ANOVA analysis carried out in achievement in mathematics,
particularly in the topic of slope, across the controlexperimental groups and
the prepost administrations.
Two Way ANOVA of Achievement in Linear
Functions (Slope)
Source 
SS 
df 
MS 
F 
P 
PrePost 
25.35 
1 
25.35 
3.17 
.078 
ControlExperimental 
34.90 
1 
34.90 
4.36 
.039 
Interaction 
49.23 
1 
49.23 
6.15 
.015 
It
is observed in Table 20 a significant effects (p < .05) in the control
vs. experimental groups and in the interaction between factors. These data confirm the significant (p
< .05) gain in achievement in slope that the experimental group had and the
slight reduction in achievement that control group exhibited at the end of the
study. A graph of the interaction
between factors showing achievement gain on slope appears in Figure 9.
Figure 9. Significant (p < .05) interaction in
achievement in linear functions: occasion X conditions (content area: slope).

Figure 10. Sample test item dealing with slope.

Item 20: Three hours after starting, car A is how many
kilometers ahead of car B? 
Figure
10 above shows a sample item from the achievement test on linear functions dealing
with slope.
The differences between means on both groups on Cartesian coordinates and related fields, a key content topic when linear functions are taught, were explored. Table 21 reports the data in the comparison between precontrol vs. preexperimental on items dealing with this topic.
Precontrol and Preexperimental Groups’ Achievement in Linear Functions (Cartesian Coordinates)
Variable 
Group 
M* 
N 
SD 
t 
df 
P 
Achievement in 
Precontrol 
1.91 
23 
1.28 
2.15 
50 
.036 
Mathematics (Cartesian Coordinates) 
Preexperimental 
1.17 
29 
1.20 



Note. *Higher mean, better achievement.
Table 21 reports a difference of means, of .74 points in favor to the control groups on the topic of Cartesian coordinates at the beginning of the treatment. The resulting t statistic points out that this difference was significant (p < .05). It may be concluded that before the treatment, control group exhibited a better performance in Cartesian coordinates and related topics such as: Cartesian plane, quadrants, and axes intercepts.
Another comparison carried out in this cluster was between pre and post administration in the control group. Table 22 summarizes the output found of this statistic test.
Pre and Post Control Group’s Achievement in Linear Functions (Cartesian Coordinates)
Variable 
Group 
M* 
N 
SD 
t 
df 
P 
Achievement in 
Precontrol 
1.91 
23 
1.28 
4.97 
22 
.000 
Mathematics (Cartesian Coordinates) 
Postcontrol 
3.22 
23 
1.31 



Note. *Higher mean, better achievement.
It
is observed in Table 22 an increase on the means of the control group from the
pretest to the posttest. The
difference between administrations was about 1.30 points. The resulting t statistic indicates that
this difference was significant (p < .05).
In
the same way, a similar comparison was carried out on the experimental
group. Table 23 reports the results.
Pre and Post Experimental Group’s Achievement
in Linear Functions (Cartesian Coordinates)
Variable 
Group 
M* 
N 
SD 
t 
df 
P 
Achievement in 
Preexperimental 
1.17 
29 
1.20 
8.29 
28 
.000 
Mathematics (Cartesian Coordinates) 
Postexperimental 
3.41 
29 
1.24 



Note. *Higher mean, better achievement.
This
comparison shows an increase in the means on achievement in linear functions
between administrations in the experimental group. The difference between means was approximately 2.24 points. The resulting t statistic reveals that this
difference is also significant (p < .05).
Although,
in these two previous comparisons, a significant difference (p < .05)
between means was found in pretest and posttest in both groups, it is also
observed that the higher difference between
means corresponds to the experimental group.
It may be concluded that in both groups there was an improvement in
linear functions, particularly in the topic of Cartesian coordinates, but in
the experimental group, this improvement was higher.
Figure
11 below shows sample items from the achievement test of linear functions
dealing with Cartesian coordinates.
Figure 11.
Sample test items dealing with Cartesian coordinates.

Item 1: The coordinates of point C are: _________ Item 2: In what quadrant is located point A?
_________ 
Graphs
The
topic corresponding to graphs constitutes another important area in the
teaching and learning of linear functions.
In this section of cluster A, the same statistical comparisons between groups
were carried out. This analysis
indicated a significant difference in only two comparisons was found. Tables 24 and 25 report the results on the
precontrol vs. preexperimental and prepost experimental comparisons,
respectively.
Precontrol and Preexperimental Groups’ Achievement in Linear Functions (Graphs)
Variable 
Group 
M* 
N 
SD 
t 
df 
P 
Achievement in 
Precontrol 
1.78 
23 
1.31 
3.51 
50 
.001 
Mathematics (Graphs) 
Preexperimental 
.69 
29 
.93 



Note. *Higher mean, better achievement.
This
comparison suggests a better achievement in graphs in favor of the control
group at the beginning of the study. The
resulting t statistic reveals that this difference was significant (p
< .05).
Pre
and Post Experimental Group’s Achievement in Linear Functions (Graphs)
Variable 
Group 
M* 
N 
SD 
t 
df 
P 
Achievement in 
Preexperimental 
.69 
29 
.93 
3.04 
28 
.005 
Mathematics (Graphs) 
Postexperimental 
1.69 
29 
1.47 



Note. *Higher mean, better achievement.
The paired samples ttest between pre and post experimental
was the other comparison where significant difference of means was found. Table 25 above, reports the results of this
test.
This
comparison illustrates an improvement in achievement in graphs in the experimental
group from the pretest to the posttest.
The tstatistic indicates that this difference of means was significant
(p < .05). These data suggest
that at the end of the treatment, students in the experimental group performed
better on the topic of linear graphs.
The difference between means for the control group was not significant (p
> .05).
Figure 12 below presents a sample item from the
achievement test dealing with the topic of graphs.
Figure 12. Sample test item dealing with graphs.

Item 14: The graph shows the distance traveled by a
tractor during a period of four hours. How fast is the tractor moving? 
Cluster B: Multiple Representations of Linear Functions
Achievement on linear functions was explored through
their four common representations: symbolic, graphical, tabular, and verbal
forms. These four comparisons between
groups were carried out in each one of these representations through
independent samples ttests, pairedsamples ttests, and ANOVA. It is important to remember that, as part of
the treatment of the study, multiple representations were strongly emphasized
in the experimental group. In the
control group, representations were just mentioned.
Symbolic Representation
The
equation, classified as formula or symbolic representation, is one of the most
widely used representations in mathematics.
In this category, a statistically significant (p < .05) difference between achievement means was
found only in the comparison prepost in the experimental group. Table 26 reports these results.
Pre and Post Experimental Group’s Achievement
in Linear Functions (Symbolic Representation)
Variable 
Group 
M* 
N 
SD 
t 
df 
P 
Achievement in 
Preexperimental 
.21 
29 
.41 
2.78 
28 
.010 
Mathematics (Symbolic Representation) 
Postexperimental 
.66 
29 
.90 



Note. *Higher mean, better achievement.
Table
26 suggests an improvement in achievement of linear function through the symbolic
representation. The difference between
means calculated was .45 points. The t
statistic indicates that this difference was significant (p <
.05). The experimental group exhibited
a gain in achievement in linear functions through the use of symbolic
representations while the control group did not. Interaction was not significant (p > .05).
Figure
13 presents a sample item from the achievement test on linear functions dealing
with symbolic representation.
Figure 13. Sample
test item dealing with symbolic representation.
The table below compares the height from
which a ball is dropped (d) and the height to which it bounces (b)

Item 7: Which equation describes this relationship? 
Graphical Representation
Graphics
constitute the representation used more often in many textbooks. Today technology, particularly computers and
calculators, has been incorporated into the curriculum to reinforce this
representation. The achievement in
linear functions through the use of this representation was explored in this
study. Table 27 includes the results of
the statistical tests carried out on this representation.
Precontrol and Preexperimental Groups’ Achievement in Linear Functions (Graphical Representation)
Variable 
Group 
M* 
N 
SD 
t 
df 
P 
Achievement in 
Precontrol 
3.13 
23 
1.79 
3.17 
50 
.003 
Mathematics (Graphical Representation) 
Preexperimental 
1.69 
29 
1.49 



Note. *Higher mean, better achievement.
Table
27 indicates a better achievement of linear functions through graphic representation
in favor of the control group at the beginning of the study. The difference between means was 1.44
points. The resulting t statistic
indicates that this difference was significant (p < .05).
Pre and Post Control Group’s Achievement in Linear Functions (Graphical Representation)
Variable 
Group 
M* 
N 
SD 
t 
df 
P 
Achievement in 
Precontrol 
3.13 
23 
1.79 
3.70 
22 
.001 
Mathematics (Graphical Representation) 
Postcontrol 
4.52 
23 
1.90 



Note. *Higher mean, better achievement.
Table 28 above shows the results on the
comparison between prepost in the control group on the items dealing with
graphical representation of linear functions.
This
change reveals an improvement in achievement in linear functions through
graphical representation in the control group.
This difference was 1.39 points and the resulting t statistic indicates
that it was significant (p < .05).
It is important to point out that since this representation is the most
shown in the textbook used in the study (Angel, 2000), and although the control
group did not receive as intensive a treatment of representations as did the
experimental group, the results show that there was an improvement in this
area.
Pre and Post Experimental Group’s Achievement
in Linear Functions (Graphical Representation)
Variable 
Group 
M* 
N 
SD 
t 
df 
P 
Achievement in 
Preexperimental 
1.69 
29 
1.49 
7.28 
28 
.000 
Mathematics (Graphical Representation) 
Postexperimental 
4.69 
29 
1.79 



Note. *Higher mean, better achievement.
Table 29 above
reports the results on achievement through graphical representation in the
prepost administration in the experimental group.
These data indicate a considerable improvement
in achievement in linear functions through graphical representation in the
experimental group. The difference between
means was 3.00 points, and the t statistic indicates that it was significant (p
< .05). Although in this same
comparison with the control group, there was also an improvement in
achievement, the difference between means reported in this data shows that in
the experimental group the difference was greater than in the control.
Table
30 reports the results of a twoway ANOVA between the pre and post
administration of the achievement test and the control and experimental groups.
Two Way ANOVA of Achievement in Linear
Functions: Graphical Representation
Source 
SS 
df 
MS 
F 
P 
PrePost 
123.67 
1 
123.67 
40.85 
.000 
ControlExperimental 
10.39 
1 
10.39 
3.43 
.067 
Interaction 
16.60 
1 
16.60 
5.48 
.021 
Table
30 suggests significant effects (p < .05) in the prepost
administration of the test. Also,
significant effects (p < .05) are observed in the interaction between
the precontrol examinations and the control and experimental groups. That is, even though both groups improved,
there is a difference in the rate of improvement in the experimental group.
Figure 14.
Significant (p < .05) interaction in achievement in linear functions:
occasions X conditions (graphical representation).

Figure
14 above shows the graph of the interactions of these two factors and the
achievement gain in graphical representations of linear functions.
Figure 15 presents a sample item from the
achievement test on linear functions dealing with graphical representation.
Figure 15. Sample test item dealing with graphical representation.
Item 23: Lisa jogs 2 miles everyday. One day after running,
she measures her pulse every two minutes. These are her results. Her pulse
rate was 140 beats per minute 2 minutes after running. It was 115 beats per
minute after 4 minutes. It was 105 beats per minute after 6 minutes. It was
90 beats per minute after 8 minutes. It was 75 beats per minute after 10
minutes. Which of these graphs best
shows her results? 
Tabular Representation
Data
summarized in tables is another form to represent linear trends. The statistical analysis carried out in this
section, reveals that significant difference on means was found in the
following comparisons: precontrol vs. preexperimental, and prepost
experimental. Tables 31 and 32
summarize the results of these two comparisons, respectively.
Precontrol and Preexperimental Groups’ Achievement in Linear Functions (Tabular Representation)
Variable 
Group 
M* 
N 
SD 
t 
df 
P 
Achievement in 
Precontrol 
1.09 
23 
.95 
3.17 
50 
.003 
Mathematics (Tabular Representation) 
Preexperimental 
.41 
29 
.57 



Note. *Higher mean, better achievement.
These
results suggest a better achievement in linear function through tabular
representation in favor of the control group.
The t statistic indicates that the difference between means was
significant (p < .05)
Pre and Post Experimental Group’s Achievement
in Linear Functions (Tabular Representation)
Variable 
Group 
M* 
N 
SD 
t 
df 
P 
Achievement in 
Preexperimental 
.41 
29 
.57 
3.62 
28 
.001 
Mathematics (Tabular Representation) 
Postexperimental 
1.07 
29 
.88 



Note. *Higher mean, better achievement.
Table
32 shows an improvement in achievement in linear equations through tabular
representation in the experimental group. The difference on means of .66 points
was significant according to the t statistic (p < .05). For the control group, the difference in means
was not significant (p > .05).
Interaction was not significant (p > .05).
Figure 16 shows a sample item from the
achievement test dealing with tabular representation of linear functions.
Figure 16. Sample test item dealing with tabular representation.
Item 24: John left his flashlight burn for 14 straight
hours. He measured the amount of
light given off (in lumens) at various times. He collected this data. Which
graph best shows his results?

Verbal Representation
Verbal
representations (such as telling a story) are not often found in college level
mathematics textbooks (Angel, 2000). The
achievement of linear functions through verbal representation was explored in
this study. The same statistical
comparisons were made in this section and the following resulted in significant
difference between means: precontrol vs. preexperimental, and prepost
experimental. Table 33 and 34 reports
the findings of these two comparisons, respectively.
Precontrol and Preexperimental Groups’ Achievement in Linear Functions (Verbal Representation)
Variable 
Group 
M* 
N 
SD 
t 
df 
P 
Achievement in 
Precontrol 
4.09 
23 
1.65 
3.74 
50 
.000 
Mathematics (Verbal Representation) 
Preexperimental 
2.00 
29 
2.24 



Note. *Higher mean, better achievement.
These
results reveal a better achievement in linear functions through verbal
representations in favor of the control group.
The difference on means of 2.09 was significant (p < .05).
Pre and Post Experimental Group’s Achievement in
Linear Functions (Verbal Representation)
Variable 
Group 
M* 
N 
SD 
t 
df 
P 
Achievement in 
Preexperimental 
2.00 
29 
2.24 
3.34 
28 
.002 
Mathematics (Verbal Representation) 
Postexperimental 
3.52 
29 
2.61 



Note. *Higher mean, better achievement.
Table
34 indicates that at the end of the study there was a gain in achievement in
linear functions through verbal representation in the experimental group. The difference on means was 1.52 points and
the t statistic reveals that it was significant (p < .05).
Table 35 reports the output from the two way
ANOVA carried out in the achievement in linear functions, particularly verbal representations
between the pre and post examination of the test, and between the control and
experimental groups.
Two Way ANOVA of Achievement in Linear
Functions: Verbal Representation
Source 
SS 
df 
MS 
F 
P 
PrePost 
6.92 
1 
6.92 
1.31 
.255 
ControlExperimental 
30.44 
1 
30.44 
5.76 
.018 
Interaction 
25.54 
1 
25.54 
4.83 
.030 
Table
35 suggests significant effects (p < .05) in the control and experimental
groups and also, in the interaction between the factors examined.
Figure 17.
Significant (p < .05) interaction in achievement in linear functions:
occasions X conditions (verbal representation).

Figure
18 shows a sample item from the achievement test on linear functions dealing
with verbal representations. For this
item, it was required that students tell a story about the situation described.
Figure 18. Sample test item dealing with verbal representation.
Item 21: Which slope is bigger? The slope of Car A or
Car B? Why? Explain your response in two or more sentences. 
Figure 18. Sample test item dealing with verbal representation.
The
previous sections of this chapter have presented the results on achievement in
mathematics, particularly in linear functions.
This important variable was explored from two perspectives: content
topics discussed in the course under study, and multiple representations of the
linear function.
The
treatment given to both groups had certain effects on the achievement in linear
functions. The statistical analyses
reports that, in general, the experimental group performed higher than the
control group, although not significant (p > .05), once the study
concluded.
This trend was also observed in the two
clusters where the achievement test was divided. In the cluster dealing with the content topics, especially in
graphs and slope, the control group performed significantly higher (p
< .05) than the experimental group at the beginning of the study. Interestingly, the experimental group
improved significantly (p < .05) in these areas at the end of the
treatment. In the topic of Cartesian coordinates,
the control group also performed significantly higher (p < .05) than
the experimental group at the beginning of the teaching experiment. Both groups got significant (p <
.05) improvements in achievement in this area once the study concluded. No significant (p > .05)
differences were found between groups at the end of the experiment.
Possible interaction effects, that is,
differences in gain scores between the control and the experimental groups were
explored using ANOVA at the clusters.
The twoway ANOVA reported that significant interactions (p <
.05) were found between factors: occasion (pre and post examinations) and
conditions (control and experimental groups) only in the content topic dealing
with slope.
In the cluster dealing with multiple
representations of the linear functions, the trend under discussion was also
observed. At the beginning of the
study, the control group showed significant (p < .05) higher
achievement than the experimental group in the following representations: graphical,
tabular, and verbal. At the end of the
study, the experimental group got a significant (p < .05) improvement
in achievement in linear functions through the symbolic, tabular, and verbal
representations. In the graphical
representation of linear functions, both groups improved significantly (p
< .05) at the end of the experiment.
From the ANOVA analysis made also in this
cluster, significant interactions (p
< .05) between factors were found in graphical and verbal
representations.
The
next chapter will include the discussion and conclusions of this study. The research questions formulated in chapter
one will be answered based on the results of this research and recommendations
will be made.