CHAPTER
5
DISCUSSION AND CONCLUSIONS
This chapter presents the answers to the four research questions proposed for this study in chapter one, based on the analyses carried out in chapter four. Conclusions, limitations of the study and recommendations follow in order to complete this chapter.
The
following research questions were investigated in this study:
1.
How did the
students in the two groups, experimental and control, compare in their prior
achievement and attitudes, and their experiences with technology?
2.
What
relationships appear to exist between attitudes and achievement in the learning
of linear functions activities?
3.
At which level
and in what ways, can the use of multiple representations be supported by
spreadsheets learning activities promote to better promote understanding of
linear functions in students at college level algebra?
4.
How well does the
medium of a powerful spreadsheet like Excel, lend itself to promoting
instruction through multiple representations?
The following sections will answer each one of
these questions separately.
Answering the First Research Question
1.
How did the
students in the two groups, experimental and control, compare in the prior
achievement and attitudes, and their experiences with technology?
Prior achievement on mathematics based on
grades reported by students from both groups was examined. The students’ achievement in mathematics is
represented by grades referred to a general knowledge in this field, not
necessarily in the topic of linear functions.
It is important to point out here that students in both groups had
completely different prior experiences, as well as, performances in
mathematics. This was attributed to the
fact of receiving the course from different schools programs (public and
private). The research question was
formulated, as the first one, since the prior achievement in mathematics was
the only variable that students exhibited before treatment for the study
started.
The results dealing with this variable,
reported on chapter four revealed no significant difference (p > .05)
between the control and experimental groups on prior achievement in
mathematics. Based on these findings,
prior achievement in mathematics based on grades seemed not to be determinant
factor in helping student attain a broad understanding about linear
functions. These results show that a
good prior achievement in mathematics did not necessarily imply a better
achievement in linear functions.
Different mathematics topics were taught in a college level algebra
course. In this study, linear functions
were strongly emphasized. According to
the results, a good performance in prior mathematics classes did not guarantee
a good performance in linear functions.
In
terms of technology use in prior mathematics courses, the students’ profiles
revealed that the use of computers, calculators, spreadsheets, Internet, among
other tools, were almost nonexistent.
Further, the profiles indicated that for the majority of the students in
the study, it was their first experience using technology, particularly,
spreadsheets, in their mathematics courses. The findings indicate that in the
experimental group, previous mathematics achievement , particularly those
supported by technology were not a decisive factor in promoting better
understanding in linear functions using spreadsheets and multiple
representations.
Answering the Second Research Question
2.
What
relationships appear to exist between attitudes and achievement in the learning
of linear functions activities?
Attitudes
toward mathematics were explored in this study by collecting data through a
scale that was administered in the beginning and at the end of the treatment,
for control and experimental groups.
The results on this variable, summarized on the previous chapter,
indicate that no significant (p > .05) was found on the comparisons
made.
The items of the attitudes scale toward
mathematics were organized in two clusters: attitudes toward technology and its
uses, and opinion or feelings toward mathematics as a subject. In terms of technology uses in mathematics
and by inspecting students’ responses on items 1 and 2 of the attitudes scale,
the experimental group had not significant (p > .05) more positive
experiences than the control group with the use of calculators to perform
routine calculations in their mathematics courses prior the study. In terms of the attitudes toward technology,
the control group exhibited a very slight (1 point of difference) positive
attitude at the beginning of the study, according to the inspection on item 3
of the scale. Once the treatment
concluded, the experimental group seemed to have a gain in positive attitudes
toward technology. These results agreed
with the treatments that each group received, as described on chapter three.
The treatment received by the experimental
group as part of the study was based on intensive use of spreadsheets while
emphasizing multiple representations of linear functions. The improvement in attitudes toward mathematics
from this group is noticeable. It could
be inferred that the traditional approach, based strictly in the use of
textbook and not technology allowed to enhance learning, did not promote an
improvement in the attitudes of the control group at the end of the study.
These
results on attitudes toward mathematics agreed with the findings on achievement
discussed in a previous section of this chapter. Based on this information, it would be sensible to conclude in
this study, that attitudes toward mathematics seemed to have limited effects on
mathematical achievement.
Answering the Third Research Question
3.
At which level
and in what ways, can the use of multiple representations be supported by the
spreadsheets learning activities to better promote understanding of linear
functions in students at college level algebra?
Since the capabilities of spreadsheets to
illustrate the multiple representations (symbolic, graphical, tabular, and
verbal) of a linear function, this technology was intensively used in this
study, particularly with the experimental group, to teach the mathematics
concepts in a college level algebra course.
The use of spreadsheets allowed students to work with multiple
representations of linear functions. As
Kaput (1992) affirms, linking one representation with the others together in
the same computer screen was essential to understanding the advantages of
technology in learning. This capability
also provided student with the understanding of how all the representations
referred to the same concept (Keller and Hirsch, 1998). Through the use of spreadsheets on the
mathematics lessons, students interacted with all representations and they
examined all the effects caused on representations when the x variable
from a linear functions assumed different values.
Mathematics
achievement, specifically on linear functions, was one of the variables
explored in this study. This variable
was measured through an achievement test, emphasizing multiple representations
administered at the beginning and at the end of the treatment. The results, reported on chapter four,
indicate that at the beginning of the study, the control group exhibited a
significant (p < .05) higher achievement than the experimental
group. It is assumed that the
experimental group’s intensive use of spreadsheets and multiple representations
in the mathematical lessons through the treatment reflected a significant (p
< .05) improvement in achievement at the end of the study.
Moreover, in order to explore students’
performance on achievement in specific content topics related to linear
functions, the research instrument on achievement was divided in a cluster
dealing with these topics. A
significant (p < .05) higher achievement was observed in favor of the
control group in all of these areas at the beginning of the study. Once the treatment concluded, no significant
(p > .05) was found between groups. It was observed in the
experimental group a significant gain (p < .05) in achievement in
mathematics in the areas of graphs and slope.
Both groups exhibited significant (p < .05) improvements in
the content of Cartesian coordinates once the teaching experiment
concluded. Interestingly, the twoway
ANOVA reported significant (p < .05) interactions between effects
(prepost administrations and groups) in the topic of slope.
In terms of the representations of the linear
function, the results revealed that in three of these representations
(symbolic, tabular, and verbal) significant differences (p < .05) were found at the end of
the study in the experimental group.
That is, students in the experimental group performed higher at the end
of this study in achievement on linear functions through symbolic, tabular, and
verbal representations. In terms of the
graphical representations, both groups exhibited a significant improvement (p
< .05) on achievement at the post administration of the test. The analysis of variance (ANOVA) carried out
between effects (groups and administrations of the test) reported significant
interactions (p < .05) in graphical and verbal representations.
The
results of this study on achievement in mathematics, through the emphasis
placed on multiple representations supported by the use of spreadsheets,
suggest that the approach presented served to promote a better understanding of
linear functions on students in a college algebra course. It was observed that students, who used
multiple representations and spreadsheets as part of their mathematics course,
performed higher in achievement in mathematics (linear functions) at the end of
the study that did the students in the control group. The data from this study suggest that the multiple
representations approach supported by technology was more successful in promote
achievement gain than the traditional approach. The findings of this research are supported by previous research
in the field of representations done by Porzio (1994). His studies revealed that the emphasis
placed in multiple representations and technology was more adequate to promote
understanding and connections between representations.
Answering the Fourth Research Question
4.
How well does the
medium of a powerful spreadsheets like Excel, lend itself to promoting
instruction through multiple representations?
Spreadsheets
were selected as the technology medium to be used in this research project
because their capabilities to promote and show the multiple representations of
the linear functions. The software
provided an engaging environment in where students explored all the
representations separately first, and then, all together in the same
worksheet. The linking process between
representations strongly recommended by Kaput (1992), Keller and Hirsch (1998),
and DufourJanvier, et al. (1987) was particularly shown on the spreadsheets.
Mathematical lessons based on spreadsheets
developed in this study, allowed students in the experimental group to explore
the effects of different values on the representations. This technology
fulfilled the objectives set for this study and supported the instructional
activities. Not only multiple
representations were supported by the use of spreadsheets, the mathematical concepts (Cartesian
coordinates, graphs, and slope) taught during the course, were introduced
through this technology. Recognized
scholars such as Fey (1989), Goldenberg (1987), Kaput (1992), and Porzio (1994)
have noticed that technology has contributed to increase the access of multiple
representations of mathematical
concepts.
In
terms of promoting instruction, the results on achievement for this study
showed that the spreadsheets approach using multiple representations was more
adequate than the traditional approach.
Furthermore, this approach based on spreadsheets seemed to serve to
enhance higher attitudes toward mathematics and technology.
Comparing the Two Approaches Used in this
Study: The Multiple Representations and the Traditional
The
purpose of this section is to present how the same mathematical concept was
taught using two different approaches.
The multiple representations, supported with spreadsheets approach used
with the experimental group and the traditional used with the control
group. The topic discussed in this
class was Cartesian coordinates.
Multiple Representation Approach
In
the class dealing with Cartesian coordinates, students in the experimental
group used a worksheet. In the first
part of this activity, six different coordinates were given and students had to
fill in the corresponding blanks the location of each one of these points. In the second part of the activity, students
offered their own coordinates, different from the presented in part one, and
satisfying certain given locations. Then, using spreadsheets they were asked to
locate the coordinates and explore the effects of points locations when the
values of x and y in (x, y) changed. To end the activity, students were
encouraged to get printouts in order to show their work. Finally, they find an equation related to these coordinates and told a story
about the application of Cartesian coordinates in their fields of study. Figures 19, 20, and 21 show the first and
the last parts of this activity done by Student 22. The answers to the questions on the worksheet appear in italic
font and printouts from spreadsheets appear in the following set of figures.
Figure 19. Worksheet sample Cartesian coordinates.
Locate in the Cartesian plane the following
coordinates. Also, determine the quadrant where the points are located.

Figure 20. Spreadsheet sample on Cartesian coordinates.
Figure 21. Sample worksheet on Cartesian coordinates.
Give an example about an equation that can
relate one of the coordinates stated above. The equation y = _{} is satisfied by the point (2,4). Because
if you substitute the values on the equation: 4 = _{} you get a true
statement: 4 = 4. Tell a story about the applications that
Cartesian coordinates may have in daily situations or another fields: I think that in medicine they serve to
imagine two cut points in a surgery.
Also, to make a map of the head where it is divided in sections. Finally, coordinates may be used to
measure distances between different bones of the body. 
Traditional Approach
This
approach was based on instructor lectures.
The only resource used in this lesson was the course textbook. Figures 22 and 23 show samples of a
student’s notes from the control group.
All notes are in Spanish.
Figure 22. Sample of student’s notes on Cartesian plane.

Figure 23. Sample of student’s notes on Cartesian coordinates.

Lessons Learned by the Researcher
Mathematics
Reasoning (MRSG 1010) is a college mathematics core course, intended primarily
for freshmen students, where technology has been slightly used throughout the semesters
that it has been taught. Although this
course has been reviewed in multiple times in terms of its objectives and how
they have been fulfilled or not, the use of more technology has been limited
and unfortunately, out of the realm of discussion. Nevertheless, regular
instructors have supported mainly the use of calculators as a supplement to
instruction. The researcher was
convinced that another kind of technology (beyond calculators) could be used in
this course to enhance achievement in mathematics. Therefore, the main challenge for the investigator of this
research was to incorporate the intensive use of computers, particularly the
use of spreadsheets to the instructional activities dealing with linear functions
of this course.
The incorporation of this technology did not
constitute an easy activity in this project.
In the beginning of the study, the students exhibited surprise and
sometimes lack of belief about how through the use of spreadsheets they could
learn the same mathematical content knowledge as did the traditional
students. Furthermore, the researcher
had to deal with the lack of expertise on students in the use of
spreadsheets. As stated earlier, this
experience using technology represented to the majority of the students their
first time using computers in a mathematics course. In order to correct this deficiency, the instructor of this study
spent some class periods teaching the basic features of spreadsheets and how
they can be used in mathematics.
In this research project, the investigator
dealt mainly with two important aspects: (a) the mathematics topics included in
the course syllabus taught during the length of the study, and (b) students
attitudes dealing particularly with the concern if spreadsheets will work or
not in order to learn linear functions and related themes. The first aspect was fulfilled when the
instructional topics were taught parallel with the traditional group and in
accordance with the syllabus. With the
purpose to fulfill the second aspect, the investigator had to motivate,
encourage and over all, show and introduce each content topic, in each class
session using spreadsheets. In this
way, students realized, once the treatment concluded, that this technology
constituted a really invaluable tool to learn mathematics.
Some Comments from Students
This
section presents some sample of comments from students in the experimental
group about their experiences using spreadsheets to learn linear functions in
mathematics. These data was collected
through a weekly electronic journal that students sent to the instructor of the
course. The electronic journal contained two questions: a) What is the big idea
that you learned in the class? and b) What topic was difficult or unclear for
you?.
During
the first week of instruction. Here
are some comments from students.
Student
27 said:
The most important thing was how do the graphs
in Excel.
Student 25 said:
The most important idea in this class was how
work with the computer and learn something new with the computer.
Student 22 said:
In my opinion, the most important thing in our
class discussion was the explanation of the spreadsheets as tool for the
course.
During
the second week of instruction. Here are some additional comments from students
in the experimental group.
Student 22 said:
The most important idea for me was the
different representations of the linear equation and how works with them based
in paper and pencil and using spreadsheets.
Student
13 said:
The Cartesian plane has been explained
perfectly. Although I realize that I
found it a little difficult to understand.
It is not that it was not well explained, but I am not skillful with
computers.
Student 26 said:
The class with the computer becomes more easy
for me that the class just explaining on the board.
Other
colleagues who were teaching the course during the semester that the study took
place, or who had taught the course before, expressed interest in knowing how
students were performing in the lessons activities based on spreadsheets. For the majority of the regular instructors
of MRSG 1010, the use of spreadsheets and multiple representations to teach
linear functions constituted something new and innovative.
As
a result of the intensive use of technology in a college level mathematics
course, the researcher’s approach to teaching changed rather dramatically and
was reinforced in terms of promoting technology use to enhance achievement in
mathematics. As the report Shaping
the Future from the National Science Foundation (George, et al., 1996)
affirms, technology should be available to all students, and they need the
opportunity to work with it, and get expertise using it as a tool of their
learning. It is reasonable to interpret
here that technology in mathematics is intended to first; to provide an
environment to promote understanding and achievement, to get help, offer
alternatives, and sometimes solutions to certain problems.
This
report discusses some barriers found in educational settings. The ineffective use of instructional
technology is one of them, pertinent to this study. This problem consists in “a specific lack of knowledge about the
hardware and technology that has been spreading into increasing use, and to
which many students are already attracted” (p. 44). Discussing on this issue, the report cited the Director of the
Science and Technology Resource Center at the Prince Georges County Community
College. She said:
I see the following as serious problems… the
challenge of teaching faculty and students how to access, utilize, and
incorporate the vast amounts of information available in print and
electronically, and learning how to utilize technology in making education more
attractive to students who might otherwise lack motivation or interest in
science, mathematics, engineering, and technology courses. (p. 44)
These problems identified in the report were
also found in this research experience.
The first situation was discussed at the beginning of this section.
Technology has been erroneously used if it
promotes misconceptions, confusion, and unclear solutions. It is important to point out here, that
certain students who participated in this study developed an excessive
dependency to the calculators or computers use to perform simple mathematics
computations and they were unable to perform them without these technological
equipments. This represents an example
about the inappropriate use and promotion of technology, where it is believed
that computers and calculators are magic boxes to solve all the problems in
mathematics. As cited in the NSF
report, Noam (1995), said: “Technology would augment, not substitute” (p. 32).
This
study through the design of mathematics lessons on linear functions, examined first,
three major variables: prior achievement in mathematics (based in reported
grades), achievement in mathematics, and attitudes toward mathematics. Second, it was compared two teaching
approaches: the multiple representation supported by technology (spreadsheets)
and the traditional.
Based
on the analyses of the data provided by the various instruments, this
researcher draws the following conclusions regarding the variables of concern.
With
respect to the prior achievement in mathematics based on reported grades, it is
concluded that this variable did not play a significant role in determining how
much mathematics was learned in the two groups, experimental and control.
Regarding
students attitudes toward mathematics, it is concluded that if positive and
higher attitudes are observed, they are a possible factor to enhance
achievement in mathematics. Students in
the experimental group had somewhat positive more attitudes toward mathematics
and performed higher in achievement.
In
regard to the achievement in mathematics, this researcher concludes that
mathematics lessons emphasizing multiple representations supported by the use
of spreadsheets constitute an appropriate teaching approach to enhance a broad
achievement on linear functions.
Students taught with the multiple representations approach achieved
higher on linear functions than students taught with the traditional
approach.
Finally,
regarding the comparison of the two teaching approaches, it was found in this
study that the approach based on multiple representations and spreadsheets is
more effective than the traditional, in promoting and enhancing achievement in
linear functions as well as more positive attitudes toward mathematics.
The
following section presents the limitations of this study. The short period of time (just four weeks of
instruction) devoted to the teaching experiment, instead the whole academic
semester, was the first limitation of this research project. Other findings could be expected if more time
was added to the treatment.
The
numbers of topics taught during this study was other limitation of this
project. Content topics strongly
related with linear functions were only emphasized throughout the study. Future studies could explore the use of
representations with other content topics discussed in a college level algebra
course.
The
researcher was the instructor of the two groups of this study: control and
experimental. Additional research in
this field can determine the effects, and other results of using more than one
teacher in similar conditions.
The
“Hawthorne effect” constituted the another limitation of the study. The changes observed at the end of the
experiment on attitudes and achievement in linear functions between control and
experimental groups could be as a result of other factors involved during the
length of the treatment, rather than the emphasis placed on multiple
representations of linear functions and the intensive use of spreadsheets. Examples of these factors could be: teacher
style, class environment, and students’ interests.
The
results of this study suggest the following recommendations. Some of these recommendations agree with the
included in the NSF report Shaping the Future (George, et. al, 1996).
To
the university and administrators:
1.
To refocus the
mission of the General Education Program, particularly the section dealing with
reasoning skills, where MRSG 1010 is part, taking into consideration students’
needs and interests.
2.
To review this
mathematics core course establishing an appropriate description in accordance
to technological advances.
To the department:
3.
To provide an
attractive curriculum in undergraduate mathematics that students can feel that mathematics
is useful in their fields of specialization.
4.
To encourage
faculty to use in the teaching of this course other technological tools, such
as, spreadsheets.
5.
To provide and
promote curricular innovations in the teaching of this course.
6.
To encourage
faculty members to do research in all mathematics courses and the publication
of their findings.
To the mathematics faculty:
7.
To believe that
all students can learn mathematics in different ways and to create and
harmonious and attractive environment that can engage students in their
learning.
8.
To suggest a
curricular review, specifying the mathematical topics that should be taught in
MRSG 1010. To recommend what content
should receive more emphasis and what should receive less.
9.
To believe that
technology has changed the education in mathematics and the use of it should
not be omitted.
10.
To promote and
encourage the use of technology (such as spreadsheets) in all mathematics
courses, not only used as supplement of instruction.
11.
To explore the
use of other teaching approaches, such as multiple representations. Using this approach, each representation of
the same concept is taught and emphasized letting students effectively manage
these representations.