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 t-Test was carried out. Table 4 reports the results.

Table 4

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: pre-control vs. pre-experimental; post-control vs. post-experimental; pre-post control; and pre-post experimental.

Students’ Opinions Toward Mathematics

Table 5 and Table 6 show the results of the first and second comparison described above, respectively.

Table 5

Pre-control and Pre-experimental 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 t-test 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 post-control vs. post-experimental. Table 6 reports the results of this analysis.

Table 6

Post-control and Post-experimental 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 t-test. 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.

Table 7

Pre and Post Attitudes Toward Mathematics: Control Group

Variable	Group	M*	N	SD	t	df	P
Attitudes Toward	Pre-control	17.35	23	2.87	.95	22	.353
Mathematics	Post-control	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.

Table 8

Pre and Post Attitudes Toward Mathematics: Experimental Group

Variable	Group	M*	N	SD	t	df	P
Attitudes Toward	Pre-experimental	16.28	29	4.10	-.652	28	.520
Mathematics	Post-experimental	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

Table 9

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	Pre-C Post-C Pre-E Post-E	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.	Pre-C	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.	Post-C Pre-E Post-E	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.	Pre-C Post-C Pre-E Post-E	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.	Pre-C Post-C Pre-E Post-E	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.	Pre-C Post-C	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.

Pre-E Post-E

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: Pre-C = Pre-Control, Post-C = Post-control, Pre-E = Pre-experimental, Post-E = Post-experimental. **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.

Table 10

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.	Pre-C Pre-E	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.	Pre-C Pre-E	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: Pre-C = Pre-control, Post-C = Post-control, Pre-E = Pre-experimental, Post-E = Post-experimental.**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.

Table 11

Students’ Use of Calculators: Summary Statistics

Statistic	Groups	Item 1	Item 2
N M Mdn SD R* Q**	Pre-control	23 2.74 3.00 1.21 4.00 2.00	23 1.91 1.00 1.31 4.00 2.00
N M Mdn SD R* Q**	Pre-experimental	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 t-test comparison between the control and experimental groups on the two items of the scale dealing with uses of technology.

Table 12

Pre-control and Pre-experimental Groups’ Uses of Technology

Variable	Group	M*	N	SD	t	df	P
Uses of	Pre-control	4.65	23	2.17	1.37	50	.176
Technology	Pre-experimental	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).

Students’ Attitudes Toward Technology

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.

Table 13

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.	Pre-C Post-C Pre-E Post-E	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: Pre-C = Pre-control, Post-C = Post-control, Pre-E = Pre-experimental, Post-E = Post-experimental. **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 twenty-five 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: pre-control vs. pre-experimental; post-control vs. post-experimental; pre-post control; and pre-post experimental. In order to examine possible interactions effects between occasions (pre and post tests) and conditions (control vs. experimental groups), two-way 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.

Table 14

Pre-control and Pre-experimental 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 post-control vs. post-experimental. Table 15 reports the results of this analysis.

Table 15

Post-control and Post-experimental 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 pre-post administration of the test in the control and experimental groups, respectively.

Table 16

Pre and Post Mathematics Achievement (Linear Functions): Control Group

Variable	Group	M*	N	SD	t	df	P
Achievement in	Pre-control	8.70	23	3.40	-1.16	22	.258
Mathematics (Linear functions)	Post-control	9.70	23	5.03

Note. Higher mean, better achievement.

The t-test shows that the difference in the means of the achievement in mathematics on the control group was not significant (p > .05)

Table 17

Pre and Post Mathematics Achievement (Linear Functions): Experimental Group

Variable	Group	M*	N	SD	t	df	P
Achievement in	Pre-experimental	4.31	29	3.67	-6.70	28	.000
Mathematics (Linear functions)	Post-experimental	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 post-test 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 two-way ANOVA (See Appendix F). The first sub-investigation deals with the clusters of items on the achievement test by content areas (Cluster A). The second sub-investigation deals with multiple representations of linear functions (Cluster B). The analyses carried out through the independent samples t-test and the paired samples t-test 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.

Cluster A: Content Topics

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: pre-control vs. pre-experimental and pre-post experimental. Tables 18 and 19 show the results of these analyses.

Table 18

Pre-control and Pre-experimental Groups’ Achievement in Linear Functions (Slope)

Variable	Group	M*	N	SD	t	df	P
Achievement in	Pre-control	5.00	23	2.30	3.79	50	.000
Mathematics (Slope)	Pre-experimental	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 pre-post comparison in the experimental group.

Table 19

Pre and Post Experimental Group’s Achievement in Linear Functions (Slope)

Variable	Group	M*	N	SD	t	df	P
Achievement in	Pre-experimental	2.45	29	2.50	-4.65	28	.000
Mathematics (Slope)	Post-experimental	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 pre-post 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 control-experimental groups and the pre-post administrations.

Table 20

Two Way ANOVA of Achievement in Linear Functions (Slope)

Source	SS	df	MS	F	P
Pre-Post	25.35	1	25.35	3.17	.078
Control-Experimental	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.

Cartesian Coordinates

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 pre-control vs. pre-experimental on items dealing with this topic.

Table 21

Pre-control and Pre-experimental Groups’ Achievement in Linear Functions (Cartesian Coordinates)

Variable	Group	M*	N	SD	t	df	P
Achievement in	Pre-control	1.91	23	1.28	2.15	50	.036
Mathematics (Cartesian Coordinates)	Pre-experimental	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.

Table 22

Pre and Post Control Group’s Achievement in Linear Functions (Cartesian Coordinates)

Variable	Group	M*	N	SD	t	df	P
Achievement in	Pre-control	1.91	23	1.28	-4.97	22	.000
Mathematics (Cartesian Coordinates)	Post-control	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 pre-test to the post-test. 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.

Table 23

Pre and Post Experimental Group’s Achievement in Linear Functions (Cartesian Coordinates)

Variable	Group	M*	N	SD	t	df	P
Achievement in	Pre-experimental	1.17	29	1.20	-8.29	28	.000
Mathematics (Cartesian Coordinates)	Post-experimental	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 pre-test and post-test 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 pre-control vs. pre-experimental and pre-post experimental comparisons, respectively.

Table 24

Pre-control and Pre-experimental Groups’ Achievement in Linear Functions (Graphs)

Variable	Group	M*	N	SD	t	df	P
Achievement in	Pre-control	1.78	23	1.31	3.51	50	.001
Mathematics (Graphs)	Pre-experimental	.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).

Table 25

Pre and Post Experimental Group’s Achievement in Linear Functions (Graphs)

Variable	Group	M*	N	SD	t	df	P
Achievement in	Pre-experimental	.69	29	.93	-3.04	28	.005
Mathematics (Graphs)	Post-experimental	1.69	29	1.47

Note. *Higher mean, better achievement.

The paired samples t-test 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 pre-test to the post-test. The t-statistic 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 t-tests, paired-samples t-tests, 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 pre-post in the experimental group. Table 26 reports these results.

Table 26

Pre and Post Experimental Group’s Achievement in Linear Functions (Symbolic Representation)

Variable	Group	M*	N	SD	t	df	P
Achievement in	Pre-experimental	.21	29	.41	-2.78	28	.010
Mathematics (Symbolic Representation)	Post-experimental	.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)

d	50	80	100	150
b	25	40	50	75

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.

Table 27

Pre-control and Pre-experimental Groups’ Achievement in Linear Functions (Graphical Representation)

Variable	Group	M*	N	SD	t	df	P
Achievement in	Pre-control	3.13	23	1.79	3.17	50	.003
Mathematics (Graphical Representation)	Pre-experimental	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).

Table 28

Pre and Post Control Group’s Achievement in Linear Functions (Graphical Representation)

Variable	Group	M*	N	SD	t	df	P
Achievement in	Pre-control	3.13	23	1.79	-3.70	22	.001
Mathematics (Graphical Representation)	Post-control	4.52	23	1.90

Note. *Higher mean, better achievement.

Table 28 above shows the results on the comparison between pre-post 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.

Table 29

Pre and Post Experimental Group’s Achievement in Linear Functions (Graphical Representation)

Variable	Group	M*	N	SD	t	df	P
Achievement in	Pre-experimental	1.69	29	1.49	-7.28	28	.000
Mathematics (Graphical Representation)	Post-experimental	4.69	29	1.79

Note. *Higher mean, better achievement.

Table 29 above reports the results on achievement through graphical representation in the pre-post 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 two-way ANOVA between the pre and post administration of the achievement test and the control and experimental groups.

Table 30

Two Way ANOVA of Achievement in Linear Functions: Graphical Representation

Source	SS	df	MS	F	P
Pre-Post	123.67	1	123.67	40.85	.000
Control-Experimental	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 pre-post administration of the test. Also, significant effects (p < .05) are observed in the interaction between the pre-control 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: pre-control vs. pre-experimental, and pre-post experimental. Tables 31 and 32 summarize the results of these two comparisons, respectively.

Table 31

Pre-control and Pre-experimental Groups’ Achievement in Linear Functions (Tabular Representation)

Variable	Group	M*	N	SD	t	df	P
Achievement in	Pre-control	1.09	23	.95	3.17	50	.003
Mathematics (Tabular Representation)	Pre-experimental	.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)

Table 32

Pre and Post Experimental Group’s Achievement in Linear Functions (Tabular Representation)

Variable	Group	M*	N	SD	t	df	P
Achievement in	Pre-experimental	.41	29	.57	-3.62	28	.001
Mathematics (Tabular Representation)	Post-experimental	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?

Time (hours)	Light given off (lumens)
0	9.5
2	8.5
3	8.5
5	6.0
8	4.2
14	0.6

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: pre-control vs. pre-experimental, and pre-post experimental. Table 33 and 34 reports the findings of these two comparisons, respectively.

Table 33

Pre-control and Pre-experimental Groups’ Achievement in Linear Functions (Verbal Representation)

Variable	Group	M*	N	SD	t	df	P
Achievement in	Pre-control	4.09	23	1.65	3.74	50	.000
Mathematics (Verbal Representation)	Pre-experimental	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).

Table 34

Pre and Post Experimental Group’s Achievement in Linear Functions (Verbal Representation)

Variable	Group	M*	N	SD	t	df	P
Achievement in	Pre-experimental	2.00	29	2.24	-3.34	28	.002
Mathematics (Verbal Representation)	Post-experimental	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.

Table 35

Two Way ANOVA of Achievement in Linear Functions: Verbal Representation

Source	SS	df	MS	F	P
Pre-Post	6.92	1	6.92	1.31	.255
Control-Experimental	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.

Summary

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 two-way 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.

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